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计量经济代写-2022S1
时间：2022-06-10
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PASS 不是我们的代言 H1 才是我们的目标
Basic Econometrics
Review Class
张帆老师
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Topic 1 The Basic Linear Model
1. Assumptions in Linear Regression
(1) Assumptions about the model
- MR1: The correct model is:
= 0 + 11 + 22 +⋯+ +
(2) Assumptions about the error term:
- MR2: Zero Conditional Expectation: The conditional expected value of the random errors is
zero:
(|) = 0
- MR3: Homoskedasticity: The variance of the random errors is constant and independent of
:
(|) =
2
- MR4: No Autocorrelation: Any pair of random errors are uncorrelated:
(, |, ) = 0 , = 1,2, …, ≠
- MR0: Not a necessary assumption: Normally Distributed Error Term
| ~ (0,
2)
• If you assume the error term is normally distributed, then you can write the model as:
= 0 + 11 + 22 +⋯+ + ℎ | ~ (0,
2), or
| ~ (0 + 11 + 22 +⋯+ ,
2)
(3) Assumptions about the Explanatory Variables: = (1, 2, … , )
- MR5a: The explanatory variables satisfy any one of the following:
• Non-stochastic Regressor: The explanatory variables are not random
• Exogeneity: The explanatory variables are random and uncorrelated with the error term
(, ) = 0 , = 1,2, …
- MR5b: No Exact Collinearity / No Perfect Multicollinearity: Any one of the X’s is not an exact
linear function of any of the other X’s.
2. The Gauss-Markov Theorem
- Under the assumptions of the linear regression model (MR1 – MR5), the OLS estimators ̂ =
(̂0, ̂1, ̂2,… , ̂) are the Best Linear Unbiased Estimators (BLUE) (have the smallest
variance) of all linear and unbiased estimators of = (0, 1, 2,… , )
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Topic 2 Statistical Properties & Inferences in Linear Model
Unit 1 Sampling Distribution of OLS Estimators
1. Sampling Distribution of OLS estimators
- If all MR1 – MR5 satisfied, the sampling distribution of OLS estimators:
~(,)
(
0
1
2
⋮
)
~
(
(
0
1
2
⋮
)
,
(
(0)
(0, 1)
(0, 2)
⋮
(0, )
(0, 1)
(1)
(1, 2)
⋮
(1, )
(0, 2)
(1, 2)
(2)
⋮
(2, )
⋯
⋯
⋯
⋱
⋯
(0, )
(1, )
(2, )
⋮
() )
)
2. Mean of OLS Estimators & Unbiasedness
- The mean of OLS estimators:
(|) = for = 1,2, …
if (|) = 0 for = 1,2, … (MAR2 holds)
- OLS estimators are unbiased estimators for population parameters, since (|) is equal
to , which is the parameter that is going to estimate
3. Variance of OLS Estimators
(1) Variance-Covariance Matrix
- The variance-covariance matrix of OLS estimators will be correct if all the following holds
• (|) = 0 for = 1,2, … (MAR2 holds)
• (|) =
2 (MR3 holds)
• (, |, ) = 0 for all , = 1,2, …, ≠ (MR4 holds)
• The
′ are non-random (MR5a holds)
- If all the conditions above hold, the formula for variance of OLS estimators:
() =
2
∑( − ̅)
2 =
2
( − 1)()
(2) Factors Affecting the Variance of the OLS Estimators
(a) Variation of Residuals (Positively Affected): 2
(b) Variation in Explanatory Variables (Negatively Affected): ∑( − ̅)
2
(c) Sample Size (Negatively Affected):
(d) Correlation between Explanatory Variables (Positively Affected): (, )
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Unit 2 Measuring Goodness of Fit in Linear Model
1. ANOVA Table in Regression & Estimator of Error Variance 2
Sum Square d.f. Mean Square
F Statistic for Overall
Significance
Model =∑(̂ − ̅)
2
=1
= /
=
/
/( − − 1)
Residual =∑( − ̂)
2
=1
− − 1
= /( − − 1)
Total =∑( − ̅)
2
=1
− 1
- The estimators of error variance:
̂2 =
∑ ̂
2
=1
( − − 1)
=
∑ ( − ̂)
2
=1
( − − 1)
=
( − − 1)
=
where
• : number of explanatory variables / + 1: number of parameters
• OLS residuals ̂ = ( − ̂) = − (0 + 11 + 22 +⋯+ )
2. Measuring Goodness of Fit in Linear Model
(1) Coefficient of Determination: 2
2 = 1 −
∑ ̂
2
=1
∑ ( − ̅)2
=1
= 1 −
= (, ̂)
- Coefficients of determination: 2 is the percentage of the variation in the dependent variable
about its mean that is explained by the regression model
- Problem of 2: 2 cannot decrease when including irrelevant variables in the model
(2) Adjusted 2: ̅2
̅2 = 1 −
×
− 1
− − 1
= 1 − [(1 − 2) ×
− 1
− − 1
]
- Interpretation of ̅2:
• ̅2 does not measure the percentage of the variation in the dependent variable explained
by the regression model
• ̅2 can be negative
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Unit 3: Hypothesis Testing
1. OLS t Test
- Test statistic:
=
−
()
~( − − 1)
where:
• + 1: number of explanatory variables in the model
• : value of stated in 0
• : OLS estimator for
• (): Standard error of , which is the square root of ()
- Rejection Rule:
0
Critical Value Method
Reject 0 if
p-value Method
Reject 0 if <
0: =
: ≠
< −(1 −
2
, − − 1)
> (1 −
2
, − − 1
= 2 × Pr( > ) for > 0
= 2 × Pr( < ) for < 0
: < < (, − − 1) = Pr( < )
: > > (1 − , − − 1) = Pr( > )
2. Type I Error and Type II Error
Reject 0 Not Reject 0
When 0 is True
Type I Error
Pr( 0| 0 ) =
Significance Level / Size of Test
Correct Decision
Pr( 0| 0 ) = 1 −
Confidence Level
When 0 is False
Correct Decision
Pr( 0| 0 ) = 1 −
Power of Test
Type II Error
Pr( 0| 0 ) =
- Type I error is the significance level we set
- Type II error is not under our control
• Probability of Type II error is unknown because the true parameter value is unknown
• Probability of Type II error can be reduced by larger sample size
- Probability of Type I error & Type II error is negatively related
• Probability of Type II error can be reduced by setting a larger
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3. The General OLS F Test
- The F Test Statistic
=
( − )/
/( − − 1)
~(, − − 1)
where:
• : Residual Sum of Square in restricted model
• : Residual Sum of Square in unrestricted model
• : Number of restrictions
• + 1: Number of parameters
- The Rejection Region
• Reject 0 if > (1 − ,, − − 1), or
• Reject 0 if = Pr( > ) <
4. F Test of Significance of Model
- The F Test Statistic
=
( − )/
/( − − 1)
=
( − )/
/( − − 1)
=
(2)/
(1 − 2)/( − − 1)
~( = , − − 1)
where:
• : Residual Sum of Square in restricted model
• / : Residual Sum of Square in unrestricted model
• : Total Sum of Square in unrestricted model
• : Number of restrictions
• + 1: Number of parameters
• 2: Coefficient of Determination of unrestricted model
- The Rejection Region
• Reject 0 if > (1 − , = , − − 1)
• Reject 0 if = Pr( > ) <
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Topic 3 Model Specification
➢ Review MR1: Model Specification: The correct model is:
= 0 + 11 + 22 +⋯+ +
• The functional form selection is correct
• No omission of relevant explanatory variables and
• No inclusion of irrelevant explanatory variables
Unit 1 Functional Form Selection
- Four Main Types of Functional Form:
Functional Form
Dependent
Variable
Explanatory
Variable
Interpretation of Slop
Coefficient
Linear-Linear Model
= 0 + 1 +
Unit form Unit form
One unit increase in is
associated with 1 unit increase
in
Log-Linear Model
log = 0 + 1 +
Logarithmic form Unit form
One unit increase in is
associated with 1 × 100%
percentage increase in
Linear-Log Model
= 0 + 1 log +
Unit form Logarithmic form
100% percentage increase in
is associated with 1 unit
increase in
Log-Log Model
log = 0 + 1 log +
Logarithmic form Logarithmic form
1% percentage increase in is
associated with 1% percentage
increase in (Elasticity)
- Additional Types of Functional Forms
(1) Polynomial
= 0 + 1 + 2
2 + 3
3 +
= 0 + 11 + 22 + 312 + 41
2 + 52
2 +
(2) Reciprocal
= 0 + 1
1
+
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Unit 2 Omission of Relevant Explanatory Variables
- Consider the correct model is:
= 0 + 1 + 2 +
and the incorrect model, which omit , is:
= 0 + 1 + with = 2 +
- The OLS estimation of the model above, which omit :
(1) = 1 + 2
(, )
()
• What we want to estimate is 1
• The actual estimate is: 1 + 2
(,)
()
• The difference: 2
(,)
()
is the biased amount
Unit 3 Inclusion of Irrelevant Explanatory Variables
- Consider the correct model is:
= 0 + 1 +
and the incorrect model, which included an irrelevant , is:
= 0 + 1 + 2 +
- Inclusion of irrelevance explanatory variable will cause:
• The OLS estimators of the parameters will be unbiased, because is an irrelevant
explanatory variable, which implies 2 = 0
• The variances of the OLS estimators in the model will be larger
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Unit 4 Testing for Model Specification
1. RESET Test for Model Specification
- Consider the following model with fitted model:
= 0 + 11 + 22 +
̂ = 0 + 11 + 22
- The RESET test model:
• Test reasonableness to include binomial terms:
= 0 + 11 + 22 + 1̂
2 +
with Hypothesis: 0: 1 = 0 against : 1 ≠ 0
• Test reasonableness to include binomial terms:
= 0 + 11 + 22 + 1̂
2 + 2̂
3 +
with Hypothesis: 0: 1 = 2 = 0 against : 1 ≠ 0 2 ≠ 0
- Conclusion:
• Reject 0 implies that original model specification is not correct
• Not reject 0 implies that original model specification is correct
- Including powers of ̂
2 can test:
• Incorrect functional form: Consider the reasonableness to include polynomial terms
• Omitted variable: Including polynomial terms may reflect omitted variables related to
included explanatory variables
• Note: RESET test cannot identify exact reason for model misspecification: functional form,
or omitted variables, or combination of two sources
2. Jarque-Bera Test for Normality of Error Term
- Null hypothesis and alternative hypothesis:
• 0: Errors are normally distributed
• : Errors are not normally distributed
- Test Statistic:
=
6
[2 +
( − 3)2
4
]~2( = 2)
where:
• : Sample skewness of OLS residuals
• : Sample kurtosis of OLS residuals
• : Sample size
- Conclusion of Jarque-Bera Test
• Reject 0: errors are not normally distributed and model specification is incorrect
• Not reject 0: errors are normally distributed and model specification is correct
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Unit 5: Collinear Economic Variables
➢ Review MR5b: No Exact Collinearity: Any one of the X’s is not an exact linear function of any
of the other X’s.
• Exact collinearity causes OLS estimation to be invalid
• Near-exact collinearity causes OLS estimation valid, but cause problems in estimation
1. Introduction to Collinearity
- Collinearity is the problem that two or more explanatory variables are highly correlated
- Consequence of collinearity
(1) OLS estimators are still unbiased, but variances and covariances of the OLS estimators
might be very large
(2) The individual coefficient t test statistic is not significance, but the model fitness is still
good, with high 2 and significant F test statistics
(3) Estimates may be sensitive to addition \ deletion of explanatory variables
- Solution to collinearity
• Exclude explanatory variables until no collinearities among explanatory variables
• However, there may be trade-off between omitted variables and collinearity
2. Test for Collinearity
(1) Pairwise Correlation
- Test the sample correlation between each pair of explanatory variables:
- For example, consider the model:
= 0 + 11 + 22 + 33 +
• The pairwise correlations: (1, 2) & (1, 3) & (2, 3)
• If there is correlation that is significantly different from 0, there should be collinearity
between suggested explanatory variables
(2) Auxiliary Regressions
- Construct regressions with one original explanatory variable to be new dependent variable,
and all the other original explanatory variables to be new explanatory variables
- For example, consider the model:
= 0 + 11 + 22 + 33 +
• The auxiliary regressions
1 = 0 + 12 + 23 +
2 = 0 + 11 + 23 +
3 = 0 + 11 + 22 +
• If the 2 is high in one or more equations above, there should be collinearity among
original explanatory variables
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Topic 5 Heteroskedasticity
Unit 1 Nature of Heteroskedasticity
1. Definition of Heteroskedasticity
- Heteroskedasticity: violation of MR3: The variance of error depends on data:
(|) =
2
• Variance of the error may depend on explanatory variables:
• Variance of the error may depend on other factors of data
- Consequences of heteroscedasticity
(1) The OLS estimator is still a linear and unbiased estimator but not BLUE
(2) The standard errors of OLS estimator are incorrect
(3) Hypothesis tests & confidence intervals will be misleading
- Solution to Heteroskedasticity:
• Use robust / White standard errors
• Use generalised least squares
Unit 2 Generalised Least Squares
- If we know the form of the heteroskedasticity, we can use generalised least squares
- In general, if we know that (|) =
2 = 2, we can transform the model
= 0 + 1 +
⇓
√
=
0
√
+
1
√
+
√
⇓
∗ = 01
∗ + 12
∗ +
∗
- New error term
∗ satisfies all the assumptions
• (
∗) = (
√
) =
1
√
() =
1
√
× 0 = 0
• (
∗) = (
√
) = (
1
√
)
2
() =
1
2 =
2
• (
∗,
∗) = (
√
,
√
) =
1
√
×
1
√
(, ) =
1
√
×
1
√
× 0 = 0
- The estimator for transformed model is called GLS Estimators, and is BLUE
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Unit 3 Feasible Generalised Least Squares
1. Feasible Generalised Least Squares (FGLS)
- FGLS can solve heteroskedasticity with unknown form
- Consider the following model:
= 0 + 11 + 22…+ +
we know is heteroskedasticity but we do not know the form
- The function form of heteroskedasticity by FGLS:
(|) =
2 × exp{0 + 11 + 22 …+ } =
2
•
′ are unknown parameters constructing the weight for GLS
• Use exponential for to ensure positive variance
• homoskedasticity when 1 = 2 = ⋯ =
2. Forms of FGLS
- Transformation of the model:
2 = 2 × exp{0 + 11 + 22…+ }
⇓
ln
2 = ln2 + 0 + 11 + 22…+
⇓
ln ̂
2 = ln 2 + 0 + 11 + 22…+ + ln ̂
2 − ln
2
⇓
ln ̂
2 = 0
∗ + 11 + 22…+ +
with
0
∗ = 0 + ln
2 and = ln ̂
2 − ln
2 = ln (
̂
2
2) = ln (
̂
)
2
but
~ ln
2(1)
(|) = −1.2707 ≠ 0
- Further Transformation
ln ̂
2 = 0
∗ + 11 + 22…+ +
⇓
ln ̂
2 = 0
∗ − 1.2707 + 11 + 22…+ + + 1.2707
⇓
= ln ̂
2 = ̃0 + 11 + 22…+ +
∗
with
̃0 = 0
∗ − 1.2707 and
∗ = + 1.2707
here
(
∗|) = 0
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3. Steps in FGLS
(1) Estimate the original model and get residuals: ̂ = ̂ −
= 0 + 11 + 22…+ +
(2) Use residuals to estimate:
= ln ̂
2 = ̃0 + 11 + 22…+ +
∗
Note:
• Estimated intercept will be off by -1.2707
• Estimated slope will be consistent (valid in large sample)
(3) Use fitted value of to run GLS:
√exp(̂)
=
0
√exp(̂)
+
11
√exp(̂)
+
22
√exp(̂)
…+
√exp(̂)
+
√exp(̂)
with
2 = (|) =
2 × exp()
(
1
√exp(̂)
) = (
1
√exp(̂)
)
2
() =
2
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Unit 4 Detecting Heteroskedasticity
1. Plot Method
(1) Plot of against with fitted line
(2) Plot of against residuals ̂
2. White’s Test
- White’s test can test heteroskedasticity with unknown form by comparing White’s standard
error & OLS standard error
- Hypothesis:
0:
2 = 2 (Homo) against :
2 ≠ 2 (Homo) against
with
•
2: White’s standard error
• 2: OLS standard error
- Test Statistics
• If the model is:
= 0 + 11 + 22 +
• The aux regression of squared residuals on all possible combinations of
′
̂
2 = 0 + 11 + 22 + 31
2 + 42
2 + 512 +
• The test statistics:
× 2~2( − 1)
with
: Sample Size
2: 2 of aux regression
: Number of parameters in aux regression
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- Problem of White’s test
• White’s test is in fact a test for model specification
• 0 assume that
• Error is homoskedasticity
• Error in independent of
′
• Functional form is correct
• Reject 0 implies that one or more conditions above not satisfied, which may or may not
identify heteroskedasticity
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Topic 6 Time Series
Unit 1 Autocorrelation
1. The Nature of Autocorrelation
- The problem of autocorrelation in error term:
• Violation of MR4: Error terms are correlated over-time (, −) ≠ 0 for some
- Consequence of autocorrelation
(4) The OLS estimator is still a linear and unbiased estimator but not BLUE
(5) The standard errors of OLS estimator are incorrect
(6) Hypothesis testing & confidence interval are invalid
- Solution to autocorrelation:
• Use HAC standard errors
• Use generalised least squares: with known form of autocorrelation
2. First-Order Autoregressive Process in Error Term: (1)
- If we know that the error term is autocorrelated one-period-ahead
(, −1) ≠ 0
we can express its form in (1) time-series regression:
= −1 +
- The error term is composed of two parts:
• −1: The carry-over from previous period
• : The correlation in
• −1 < < 1: The carry-over will diminish period by period
• : The new shock this is uncorrelated overtime, and satisfies Gauss-Markov Theorem
- Properties of First-Order Autoregressive Errors:
• Zero mean
() = 0
• Homoscedasticity
() =
2 =
2
1 − 2
• Covariance between successive error terms:
(, −) =
2 =
2
1 − 2
• Correlation between successive error terms:
(, −) =
(, −)
√()√(−)
=
2
√2√2
=
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3. Generalised Least Square
(1) Methods of Generalised Least Square
- The time series regression model:
= 0 + 1 + with = −1 +
with properties
• () = 0
• () =
2 =
2
1−2
• (, −) =
2 =
2
1−2
- Substitute the (1) error into the model:
= 0 + 1 +
= 0 + 1 + (−1 + )
−1 = −1 − 0 − 1−1⟸ −1 = 0 + 1−1 + −1
⟹ = 0 + 1 + (−1 − 0 − 1−1) +
− −1 = 0(1 − ) + 1( − −1) +
- The transformed model:
∗ = 0
∗ + 1
∗
∗ +
with
∗ = − −1; 0
∗ = 0(1 − ); 1
∗ = 1;
∗ = − −1
where
• in the new model satisfies Gauss-Markov Theorem
• (T-1) observations are included in new model
(2) Steps in Generalised Least Square
➢ Step 1: Estimate by OLS
- Since = −1 + , we ca estimate using OLS residuals as estimation of error term
• Obtain the OLS residuals
̂ = − 0 − 1
• Put OLS residuals into (1) model
̂ = ̂−1 +
• Estimation of
̂ =
∑ ̂̂−1
=2
∑ ̂−1
2
=2
➢ Step 2: Implement GLS
- The transformed model:
∗ = 0
∗ + 1
∗
∗ +
with
•
∗ = − −1;
• 0
∗ = 0(1 − );
•
∗ = − −1
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4. Test for Autocorrelation – LM Test
- Consider the following time series model:
= 0 + 1 +
where we want to test whether: (, −) ≠ 0 for some ≥ 1
- LM Test for (1) errors: = −1 +
• Hypothesis: 0: = 0 against 1: ≠ 0
• Test Statistics: Using 2 of the following model
̂ = 0 + 1 + ̂−1 +
= × 2~2(1)
• Reject 0 if >
2(, 1)
- LM Test for (2) errors: = 1−1 + 2−2 +
• Hypothesis: 0: 1 = 2 = 0 against 1: At least one ≠ 0, = 1,2
• Test Statistics: Using 2 of the following model
̂ = 0 + 1 + 1̂−1 + 2̂−2 +
= × 2~2(2)
• Reject 0 if >
2(, 2)
- Test for () errors: = 1−1 + 2−2 +⋯+ − +
• Hypothesis: 0: 1 = 2 = ⋯ = = 0 against 1: At least one ≠ 0, = 1,2…
• Test Statistics: Using 2 of the following model
̂ = 0 + 1 + 1̂−1 + 2̂−2 +⋯+ ̂− +
= × 2~2()
• Reject 0 if >
2(, )
- Problem of LM Test
• LM tests need us to know the form of autocorrelation
• If we do not know the form of autocorrelation:
• Testing Down Approach
• Testing up a pp roach
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Unit 2 Non-Stationary Time Series
- In time series models, all time series data included should be stationary
- Summary: Time Series Patterns
Name Plot Stationarity Unit Root ADF Test Type
Stationary
Time
Series
/
Mean-
Reverting
Stationary No Drift
Trend
Stationary
Non-Stationary No Drift
Random
Walk
Non-Stationary Yes Trend
Random
Walk with
Drift
Non-Stationary Yes Trend
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1. Stationary Time Series
(1) Definition of Stationary
- A time series data is stationary if:
• Its mean is constant overtime
() =
• Its variance is constant overtime
() =
2
• Its autocovariance for any given lag length is constant overtime
(, −) = (, +)
(2) Stationary Time Series in (1) Model
- The (1) model of which is stationary:
= −1 + with || < 1
where satisfies Gauss-Markov Theory
- Then, we have:
• () = 0
• () =
2 (
1
1−2
)
• (, +) =
2 (
1−2
)
(3) Different Kinds of (1) Model with Stationary Time Series
- (1) Model with zero mean
= −1 +
- (1) Model with non-zero mean
= + −1 + with () = =
1−
• De-meaning:
= + −1 +
= (1 − ) + −1 +
− = (−1 − ) +
with = (1 − )
- (1) Model with non-zero mean & linear trend (Trend Stationary which is Non-Stationary)
= + + −1 + with () = +
• De-meaning & de-treading:
= + + −1 +
= [(1 − ) + ] + (1 − )
− − = [−1 − − ( − 1)] +
with = (1 − ) + and = (1 − )
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2. Non-Stationary Time Series
- In the (1) model of , if = 1, then is non-stationary
(1) Random Walk Model
= −1 +
• Apply recursive substitution
= 0 +∑
−1
=0
then
() = (0) = 0
() = (0 +∑
−1
=0
) = 2
• Random walk is non-stationary because its variance increases over time
(2) Random Walk with Drift
= + −1 +
• Apply recursive substitution
= + 0 +∑
−1
=0
then
() = + 0
() =
2
• Random walk is non-stationary because its mean and variance increases over time
3. Spurious Regression
- Definition of Spurious Regression
• If we regress two non-stationary time series on each other, we will get a significant OLS
coefficient even if the series are unrelated
• Consider the following two random walk time series
= −1 + 1
= −1 + 2
and are uncorrelated, however in the following model:
= 0 + 1 +
estimate of 1 is usually statistically significant suggesting and are correlated
- Consequence of Spurious Regression
• OLS estimators’ statistical properties are not valid
• OLS estimators’ statistical inferences (hypothesis testing) are not correct
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4. Testing for Non-Stationarity: Dickey Fuller (DF) Test
(1) Dickey Fuller (DF) Test
- Consider the model:
= −1 +
or
Δ = ( − 1)−1 +
= −1 +
- Hypothesis:
0: Non-Stationary / = 1 / = 0
0: Stationary / < 1 / < 0
- Different Kinds of Models
• DF Test I: If time series appears to fluctuate around mean zero:
Δ = −1 + with 0: = 0
• DF Test II: If time series appears to fluctuate around non-zero mean:
Δ = + −1 + with 0: = 0
• DF Test III: If time series appears to fluctuate around linear trend:
Δ = + + −1 + with 0: = 0
(2) Augmented Dickey Fuller (ADF) Test
- ADF test consider possible autocorrelation in error term
- We add augmented terms (lagged first difference terms in ) to eliminate autocorrelation
Δ = + −1 + ∑ Δ−
=1 + with 0: = 0
(3) Test Statistics
- The test statistics for DF test & ADF test follows DF distribution, not t distribution
- We just focus on p-value of the tests
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Topic 7 Stochastic Regressors
Unit 1: Stochastic Regressors and Problem of Endogeneity
1. Definition of Endogeneity
- MR5a version 2: Exogeneity requires that the explanatory variables are random and
uncorrelated with the error term:
( , ) = 0
- Endogeneity is a problem that explanatory variable is correlated with error term:
( , ) ≠ 0
2. Consequence of Endogeneity
- OLS estimators are biased and inconsistent
- OLS standard errors are incorrect
- Hypothesis tests & confidence interval are invalid
3. Sources of Endogeneity
(1) Classical Measurement Error
(2) Omitted Variables
(3) Lagged Dependent Variable with (1) Error Structure
(4) Simultaneous Causal Equations
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Unit 2: Sources of Endogeneity
1. Classical Measurement Error
- Classical measurement error arises when explanatory variables are measured with error
- Consequences of Classical Measurement Error
• OLS estimators are biased, and standard errors are incorrect
• Attenuation Bias: 1 is biased towards zero
• No bias if and only if no measurement error
2. Omitted Variables
- Omitted variable bias arise if
• One or more relevant explanatory variables are omitted from the true mode, and
• Omitted explanatory variables are correlated with included explanatory variables
- Consider the correct model is:
= 0 + 1 + 2 +
with
• (,) = 0
• (, ) = 0
- Consider the incorrect model, which omit :
= 0 + 1 + (2 + )
= 0 + 1 +
- In this estimated model:
(, ) = (, 2 + )
= 2(,) + ( , )
= 2(,)
Endogeneity arises if 2 ≠ 0 and (,) ≠ 0
- The incorrect OLS estimation:
(1) = 1 + 2
(,)
()
The bias:
= 2
(,)
()
3. Lagged Dependent Variables
- In time series model, when including a lag of dependent variable as an explanatory variable:
= 0 + 1−1 +
- If there is autocorrelation in error term
• correlated with −1
• −1 correlated with −1: endogeneity
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Unit 3: Instrumental Variable (IV) and Two Stage Least Squares (2SLS)
- Instrumental Variable (IV) and Two Stage Least Squares (2SLS) is the effective and efficient
method to solve all sources of endogeneity
1. Instrumental Variable Validity Condition
- Consider the model where is endogenous
= 0 + 1 +
- Consider the instrumental variable , that satisfies IV validity conditions
(a) Exclusive Restriction
• does not directly correlated with dependent variable
• does not correlated with error term: (, ) = 0
(b) Instrumental Relevance
• is correlated with endogenous variable ( , ) ≠ 0
2. Two Stage Least Square
- Consider the model where is endogenous and is a valid IV for
= 0 + 1 +
- The process of Two Stage Least Squares (2SLS) estimation:
(a) Stage 1: Regress on to get a fitted value of :
̂ = ̂0 + ̂1
(b) Stage 2: Estimate original model, replacing with fitted value of
̂ = 0 + 1̂
- By the process of 2SLS, the estimator 1 is called 2SLS estimator or IV estimator
3. Properties of IV Estimators
(1) IV estimators are consistent if IV validity conditions satisfied
(2) In small sample, IV estimators are biased
(3) In large sample, IV estimators are approximately unbiased and normally distributed
- In the model:
= 0 + 1 +
where is endogenous and is a valid IV for
- The sampling distribution of IV estimator in large sample
1
. ~ (1,
2
2
2 )
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(4) The Variance of IV Estimator
- In large sample, IV estimators have variance:
(1
) =
2
2
2
where
•
2: Population variance of
•
2: Population variance of
•
2 : Square of population correlation between and
- Estimates of Variance of IV Estimator
̂(1
) =
̂
2
2
2 =
̂
2
∑( − ̅)2
2
where
• ̂
2 : Sample variance of residuals
•
2: Sample variance of
•
2 : Square of sample correlation between and
- Properties of Variance of IV Estimator
(a) Suppose (, ) = 0, then for the OLS estimator & IV estimator
̂(1
) =
̂
2
∑( − ̅)2
=
2 ̂(1
)
When OLS is valid, OLS estimator will have a lower variance than IV estimator
(b) Higher sample correlation
2 lowers the estimated variance of IV estimator
(c) Larger sample size lower variance of IV estimator
(d) Greater variance in lower variance of IV estimator
4. Weak Instruments
- Weak Instrument: (, ) ≈ 0
- How to detect weak instrument: Rule of thumb: F test statistic of IV in stage 1 less than 10
- Problem of weak instrument:
• Normal distribution provides poor approximation to sampling distribution of the IV
estimator in large sample
• No justification for using t-tests and F-tests which are based on assumption of normality
• Statistical inference: affects conclusions from hypothesis tests, which are based upon an
incorrect distribution of the test statistic when 0 is true
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Unit 4: Simultaneous Causal Equation System
1. Definition of Simultaneous Causal Equations
- Simultaneous Causal Equations: Two variables are jointly determined
- Simultaneous causal equations is one source of endogeneity: (, )
2. Simultaneous Causal Equations: Structural Equations
- Consider two simultaneous caused variables: 1 & 2
{
1 = 12 + 11 + 1 (1)
2 = 21 + 22 + 2 (2)
(1) Endogenous Variables
- In the equations system above, endogenous variables:
• 2 is the endogenous variable in equation 1
• 1 is the endogenous variable in equation 2
• 1 & 2 are both the endogenous variables in the system
(2) Exogenous Variables & Instrumental Variables
- In simultaneous causal equation system, all exogenous variables in the system can be
considered as instrumental variables, irrespective of which equation they are in
- In the equations system above, exogenous variables:
• 1 & 2 are the exogenous variables / instrumental variables in the system
• 1 is the exogenous variable in only equation 1
• 1 is used to identify equation 1
• 1 can be used as exclusive IV for 1
• 2 is the exogenous variable in only equation 2
• 2 is used to identify equation 2
• 2 can be used as exclusive IV for 2
(3) Identification
- In simultaneous equation system, an equation is identified if it contains exogenous variable
that does not exist in other equations
- In a system of 2 simultaneous equations with 2 endogenous variables, at least 1 equations
must be identified; we need at least 1 exogenous variables
- If a simultaneous equation system is identified, the parameters in this system can be
estimated consistently
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3. Simultaneous Causal Equations: Reduced Forms
(1) Definition and Illustration of Reduced Forms
- The reduced forms express each endogenous variable in terms of all exogenous variables in
the system
- Consider the structural equations:
{
1 = 12 + 11 + 1 (1)
2 = 21 + 22 + 2 (2)
assuming:
• (1) = (2) = 0
• (1, 2) = 0
- Reduced forms:
{
1 = 111 + 122 + 1 (1)
2 = 211 + 222 + 2 (2)
where coefficients ′ are functions of structural coefficients
(2) Properties of Reduced Form
- Reduced form can be estimated by OLS, because no endogeneity in reduced form
- Reduced form is actually the stage 1 of 2SLS
- Reduced form parameters are non-linear function of structural parameters
• We cannot estimate structural parameters
• We can estimate non-linear function of structural parameters
4. Two-Stage Least Square
(1) 2SLS in Simultaneous System with Two Equations
- Consider the structural equations
{
1 = 12 + 11 + 1 (1)
2 = 21 + 22 + 2 (2)
(a) Estimation of Equation 1
- Endogenous Variable: 2
- Instrumental Variable of 2: 2, with IV validity conditions
• Exclusion Restriction: (2, 1) = 0: 2 won’t directly correlated with 1
• Instrumental Relevance: (2 , 2) ≠ 0: 2 correlated with 2
- Two Stage Least Square
• Stage 1: Regress 2 on all IVs in the system and obtain fitted value
̂2 = ̂211 + ̂222
• Stage 2: Estimate equation (1), replacing 2 with fitted value
̂1 = ̂1̂2 + ̂11
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(b) Estimation of Equation 2
- Endogenous Variable: 1
- Instrumental Variable of 1: 1, with IV validity conditions
• Exclusion Restriction: (1, 2) = 0: 1 won’t directly correlated with 2
• Instrumental Relevance: (1, 1) ≠ 0: 1 correlated with 1
- Two Stage Least Square
• Stage 1: Regress 1 on all IVs in the system and obtain fitted value
̂1 = ̂111 + ̂122
• Stage 2: Estimate equation (2), replacing 1 with fitted value
̂2 = ̂2̂1 + ̂22
(2) Properties of Two Stage Least Square
• The 2SLS estimator is biased but consistent
• In large samples, 2SLS estimator is approximately normally distributed
• The variances and covariances of the 2SLS estimator are unknown in small samples but
can be approximated in large sample
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Topic 8 Panel Data Model
Unit 1: Panel Data Concepts
1. Introduction to of Panel Data
- Panel data is the combination of cross-sectional data & time-series data
• N: Number of cross-sectional units
• T: Number of time periods
2. Variables in Panel Data Frame
- : Time-variant & Individual-variant
- : Time-invariant & Individual-variant
- : Time-variant & Individual-invariant
- : Time-invariant & Individual-invariant
3. Estimation of Panel Data Model
- Consider a general panel data regression:
= 0 + 11 + 22…+ +
The purpose is to estimate that reveals relationship between & that is applicable
across all individuals & across all time periods
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Unit 2: Pooled OLS Model
1. Definition of Pooled OLS
- Pooled model: Simple OLS applied on whole panel data set
- Pooled OLS model example:
= 0 + 11 + 22 +
where
• : Individual Index: = 1,2, …
• : Time Index: = 1,2, …
• Total observations: ×
2. Assumptions of Pooled OLS
- Required assumptions for pooled OLS estimator to be consistent
- Assumptions on individual dimension:
• (|) = 0 Zero Mean
• (|) =
2 Homoskedasticity across individuals
• (, ) = 0 Exogeneity
• (, ) = 0 No autocorrelation across individuals
- Assumptions on time dimension:
• (, |) = ≠ 0 Autocorrelation across time
• (, |) = () = Heteroskedasticity across time
3. Problem of Heteroskedasticity & Autocorrelation across Time
- Consequence of Pooled OLS
• The OLS estimator is still unbiased & consistent but not BLUE
• The standard errors are incorrect
• The hypothesis testing & confidence interval based on these standard errors are invalid
- Solution: use cluster-robust standard errors
• Clusters: the time series observations on individuals
• Cluster-robust standard errors are only valid in large sample
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Unit 3: Fixed Effects Model
1. Fixed Effects Model
- Fixed effect model treats individual time-invariant characteristics as a constant parameter
- Fixed effects model:
= 0 + 11 + 22…+ +
2. Fixed Effect Estimation: Dummy Variable Method
- Fixed effects model by individual dummy variables:
= 0 + 11 + 22…+ +
where
= {
1 for individual j
0 othervise
- Required assumptions for OLS estimator to be consistent
• (|) = 0 Zero Mean
• (|) =
2 Homoskedasticity
• (, |) = 0 No autocorrelation
• (, ) = 0 Exogeneity
- As sample size increase:
• Estimation of 1, 2, … , are consistent
• Estimation of 0
′ are inconsistent
3. Fixed Effect Estimation: Within-Transformation
- Within-transformation method keeps only slope coefficients and remove individual intercepts
- Process of within-transformation
• Fixed effect model:
= 0 + 11 + 22…+ + (1)
• Take time average on each term:
1
∑
=1
=
1
∑0
=1
+
1
∑11
=1
+
1
∑22
=1
…+
1
∑
=1
+
1
∑
=1
̅ = 0 + 1̅1 + 2̅2…+ ̅ + ̅ (2)
• Take (1) – (2):
− ̅ = 1(1 − ̅1) + 2(2 − ̅2) + ⋯+ ( − ̅) + ( − ̅)
∗ = 11
∗ + 22
∗ …+
∗ +
∗
- The final version is the within-transformed model:
∗ = 11
∗ + 22
∗ …+
∗ +
∗
- Estimation of within-transformed model gives within estimator
• Individual intercepts are removed
• Time-invariant variables will also be removed
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Unit 4: Random Effect Model
1. Definition of Random Effect Model
- Random effect model treats individual time-invariant characteristics as a random variable
- Random effect model
= + 11 + 22…+ +
- Assumptions about the model:
• (|, ) = 0 Zero Mean
• (|, ) =
2 Homoskedasticity
• (, |, ) = 0 ≠ No autocorrelation across time
• (, |, ) = 0 ≠ ≠ No autocorrelation across individual
• (, |) = 0 Exogeneity
- Assumptions about the individual random effect :
• (, ) = 0: & are independent
• (|) = 0: Expectation of is the intercept
• (|) =
2: Constant Variance
• ( , ) = 0 ≠ : No correlation among individuals
• (, ) = 0: Uncorrelated with error term
2. Random Effect Model Estimation: Generalised Least Squares
(1) Transform the Random Effect Model
- Consider the following random effect model:
= + 11 + 22 +
- Add & Subtract 0:
= 0 − 0 + + 11 + 22 +
= 0 + 11 + 22 + ( − 0 + )
= 0 + 11 + 22 +
- In the final version:
= 0 + 11 + 22 +
• Mean & Variance of Error Term:
(|, ) = (|) − 0 + (|, ) = 0 − 0 = 0
(|, ) = () + () + 2(, ) =
2 +
2
• Autocovariance of Error Term:
(, ) = 0: No autocorrelation across individual
(, ) =
2: Autocorrelation across time
(, ) = 0: No autocorrelation across individual & across time
• Autocorrelation of Error Term across Time:
= (, ) =
(, )
√()()
=
2
2 + 2
Interpretation: proportion of total variance in the error term that can be attributed
to variance in the individual time-invariant components
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(2) GLS Estimation of Random Effects Model
- Transformed random effects model:
= 0 + 11 + 22 +
Define:
= 1 −
2
√2 + 2
Then estimate transformed model to obtain estimation of ̂
2 & ̂
2 and calculate ̂
- The GLS mode to be estimated:
− ̂̅ = (1 − ̂)0 + 1(1 − ̂̅1) + 2(2 − ̂̅2) + ( − ̂̅)
∗ = 0
∗ + 11
∗ + 22
∗ +
∗
- Properties of Random Effects Estimators (GLS)
• Uses both within individual and between individual variation to identify coefficients
• Could estimate time-invariant variables
• GLS estimator generally provides estimates with lower standard errors
3. Correlated Random Effects (CRE)
- Consistency of Random Effects Estimators:
• The random effects estimator will be consistent if the random error = +
uncorrelated with any of the explanatory variables
• If this is not satisfied, then the random effects estimator is biased and inconsistent
- Consider the following original random effect model
= 11 + 22 + +
with correlated with explanatory variables:
= + 1̅1 + 2̅2 +
then substitute in the original model
= + 11 + 22 + 1̅1 + 2̅2 + +
then estimate this model by random effect estimator
- The CRE estimators of 1 & 2:
• Will be identical to fixed effect estimator if 1 & 2 is not zero
• Will be identical to random effect estimator if 1 & 2 is zero
• We can test this by 0: 1 = 2 = 0
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Unit 5: Comparisons among Panel Data Models
1. Comparison of Models:
Pooled OLS Fixed Effects Random Effects
Time-invariant
Unobserved
Heterogeneity
Not Allow Allow
Treat as constant coefficient
Correlated with
Allow
Treat as random variable
Uncorrelated with
Autocorrelation across
Time
May exit
Use cluster-robust SE
May exit
Use cluster-robust SE
Must exit
Generalised Leased
Squares
Identification of
Variation
Within individual variation Within individual
variation & Between
individual variation
Variables Included Not allow Allow
2. Choose between Models
(1) Pooled OLS Model vs Fixed Effect Model
- Fixed effects model by individual dummy variables:
= 0 + 11 + 22…+ +
- Hypothesis:
• 0: Fixed effects 0
′ are jointly zero
• 1: At least one fixed effect 0
′ are not zero
- Test Statistic: F Test
- Conclusion:
• Reject 0: Fixed effects is the appropriate model
• Not Reject 0: Pooled OLS is the appropriate model
(2) Pooled OLS Model vs Random Effect Model
- The variance of random effect: () =
2
- Hypothesis:
• 0:
2 = 0
• 1:
2 ≠ 0
- Test Statistic: 2(1)
- Conclusion:
• Reject 0: Random effects is the appropriate model
• Not Reject 0: Pooled OLS is the appropriate model
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(3) Fixed Effect Model vs Random Effect Model
- FE Estimator & RE Estimator
• When ( , ) = 0: Both FE & RE consistent but RE has lower variance
• When ( , ) ≠ 0: Only FE consistent
- Hausman Test: Compare difference between FE Estimator & RE Estimator
- Hypothesis or one pair of parameters:
• 0: , = ,
• 1: , ≠ ,
- Test Statistic for each pair of parameters:
=
, − ,
√(, − ,)
Combined test statistic asymptotically follows 2()
- Conclusion:
• Reject 0: Fixed effects is the appropriate model
• Not Reject 0: Random effects is the appropriate model
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Topic 9 Count Data Model
1. Model for Count Variable:
- Dependent variable is a count variable that take non-negative integer values: 0,1,2,…
- Characteristics of Count Data:
• Outcome has no natural upper bound
• Outcome will likely be zero for a non-trivial proportion of the sample
• Outcome is often right skewed
2. The Poisson Regression Model
- Poisson distribution can explain and predict the probability of count variable
- The Poisson parameter:
= exp(0 + 11 + 22…+ )
- The dependent variable follows Poisson distribution on parameter :
| ~ ()
with probability mass function
Pr( = |) = ( = |) =
− ×
!
- Expected Value & Variance
• (|) =
• (|) = This is the intrinsic heteroskedasticity: solve by robust standard error
3. Estimation of Poisson Model
- Use maximum likelihood estimation (MLE), rather than OLS, to estimate Poisson Model
- The Likelihood Function: represents the probability of observing the data:
(0, 1, … ) = (1, 2, … ) = (1)(2)…()
=
−1 × 1
1
1!
×
−2 × 2
2
2!
…×
− ×
!
- The MLE choose estimators of (0, 1, …) that can maximize the Likelihood Function
- Properties of MLE estimator in large sample:
• Normally distributed
• Consistent
• Best: has the smallest variance
4. Prediction of Poisson Model
- Prediction of Conditional Mean:
(|)̂ = ̂ = exp(̂0 + ̂11 + ̂22…+ ̂)
- Prediction of Probability
Pr( = |)̂ =
−̂ × ̂
!
ℎ = 0,1,2, …
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5. Marginal Effects in Poisson Model
(1) Continuous Variable:
- Marginal Effects of : Semi-Elasticity
%∆(|)
∆
= 100 × [
1
(|)
×
(|)
] = 100 ×
Interpretation: One unit increase in is expected to cause × 100% percentage
increase in average number of
- Marginal Effects of ln : Elasticity
%∆(|)
∆ ln
≈
%∆(|)
%∆
=
Interpretation: One percent increase in is expected to cause % percentage increase
in average number of
(2) Dummy Variable
- Marginal Effect of as Dummy Variable
%∆(|)
∆
≈ 100 × [
1
(|)
×
(|)
] = 100 × ( − 1)
Interpretation: Moving from = 0 to = 1 is expected to cause (
− 1) × 100%
percentage increase in average number of
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Topic 10 Binary Outcome Model
Unit 1: Introduction to Binary Outcome Model
- Dependent variable is a Bernoulli variable (dummy variable) that take either 0 or 1
- Distribution of :
= {
1 with probability
0 with probability 1 −
with pmf
() =
× (1 − )
(1−) = {
ℎ = 1
1 − ℎ = 0
mean & expectation
• () =
• () = (1 − ) Intrinsic Heteroskedasticity
- The fitted value ̂: probability
Unit 2: Linear Probability Model (LPM)
1. Introduction to Linear Probability Model
- Regression Model:
= 0 + 11 + 22…+ +
- Conditional Mean Function:
(|) = = 0 + 11 + 22…+
- Random Error: Bernoulli Distributed: Not Normally Distributed
= − (|) = {
1 − (0 + 11 + 22…+ ) ℎ = 1 ℎ
−(0 + 11 + 22 …+ ) ℎ = 0 ℎ 1 −
2. Heteroskedasticity in LPM
(1) Heteroskedasticity
- The variance of error term in LPM:
(|) = (1 − ) =
2
- The form of heteroskedasticity is in a quadratic manner with
- Consequence of heteroskedasticity:
• Standard errors for OLS estimator are incorrect
• Confidence interval & hypothesis testing are misleading
- Solution:
• Huber-White / Robust Standard Error
• Feasible Generalised Leased Square
(2) Feasible Generalised Leased Square
- Steps of FGLS for LPM:
• Estimate LPM by OLS: = 0 + 11 + 22…+ +
• Calculate ̂ = ̂0 + ̂11 + ̂22…+ ̂
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• Estimate the weighted model:
√(1 − )
=
0
√(1 − )
+
11
√(1 − )
+
22
√(1 − )
…+
√(1 − )
+
√(1 − )
- Problem of FGLS
• When ̂ = 0 or ̂ = 1: weighting series is undefined
• When ̂ outside range [0,1]: weighting series is undefined
3. Marginal Effect of LPM
- Marginal Effect
=
(|)
=
Pr( = 1|)
- Interpretation:
• Continuous Variable: One unit increase in cause increase in probability for = 1
• Dummy Variable: Moving from = 0 to 1 cause increase in probability for = 1
• Intercept: Average probability for = 1 is
4. 2 of LPM
- 2 in LPM is high if observations are clustered around mean of (low variance of )
- Trade-off:
• High 2 ⟹ Low variance of ⟹ Low precise of estimate
• Low 2 ⟹ High variance of ⟹ High precise of estimate
5. Problem of LPM
- Predicted value ̂ may outside range [0,1] and lead to a negative predicted variance
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Unit 3: Probit Model
1. Introduction to Probit Model
- Probit Model: Fit data in cumulative distribution function of standard normal distribution
Pr( ≤ 0 + 11 + 22…+ )
= Φ(0 + 11 + 22 …+ )
- Slope of Φ(0 + 11 + 22…+ ) is not constant, which increase rapidly then slowly
2. Latent Variable Foundations for Probit Model
- Observed Dependent Variable
= {
1 ′′
0 ′′
- Latent Variable:
∗: Linear Model:
∗ = 0 + 11 + 22…+ + with | ~ (0,
2)
•
∗ ∈ (−∞,+∞) which can take any real value
•
∗ is unobserved
- Relationship between Observed Dependent Variable & Latent Variable:
= {
1
∗ > 0
0
∗ ≤ 0
Pr( = 1) = Pr(
∗ > 0) = Φ(
0 + 11 + 22…+
)
Pr( = 0) = Pr(
∗ ≤ 0) = 1 − Φ(
0 + 11 + 22…+
)
• Index Function: 0 + 11 + 22…+
• Index function is large & positive: Φ(. ) → 1
• Index function is large & negative: Φ(. ) → 0
• Index function is zero: Φ(. ) = 0.5
• Interpretation of
∗: A possible dependent variable of which is continuous
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- Probit Model is a link between Normal Distribution & Bernoulli Distribution
• The distribution of is Bernoulli Distribution
= {
1 with probability
0 with probability 1 −
where
= Φ(
0 + 11 + 22…+
)
• The pmf of :
() =
× (1 − )
(1−) = {
ℎ = 1
1 − ℎ = 0
3. Estimation of Probit Model: Maximum Likelihood Estimation (MLE)
- Maximum Likelihood Estimation (MLE) choose (0, 1, … ) that maximize
• Likelihood Function:
(0, 1, … ) = (1, 2, … ) = (1)(2)…()
• log Likelihood Function:
ln (0, 1, … ) = ln (1, 2, … ) = ln (1) + ln (2)…+ ln ()
where
() = {
Φ(
0 + 11 + 22…+
) ℎ = 1
1 − Φ(
0 + 11 + 22…+
) ℎ = 0
- Properties of MLE estimator in large sample:
• Normally distributed
• Consistent
• Best: has the smallest variance
- Note: MLE in R
• Coefficients reported are divided by : (
̂0
), (
̂1
), …, (
̂
)
• Reports Z-statistic rather than t-statistic, because of large sample
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4. Truth Table
- After estimation, we generate predicted model
̂ = Φ(
0
+
1
1 +
2
2…+
)
- Then predicted dependent variable:
̂ = {
1 ̂ ≥
0 ̂ <
This decision may lead to correct or incorrect prediction
- Truth Table:
Number of Values
Truth
= 1 = 0
Prediction
̂ = 1
a
predicted correctly for = 1
c
predicted incorrectly for = 0
̂ = 0
b
predicted incorrectly for = 1
d
predicted correctly for = 0
• Proportion of correct prediction of = 1:
+
• Proportion of correct prediction of = 0:
+
5. Marginal Effect of Probit Model
(1) Continuous Variable
- Marginal Effect of as continuous variable
Pr( = 1)
=
Φ(
)
(
)
×
(
)
= (
0
+
1
1 +
2
2…+
) × (
)
- Calculation of ME
(a) Calculate predicted value of index
̂
∗ = (
̂
) =
0
+
1
1 +
2
2 …+
(b) Calculate estimated marginal effect
Pr( = 1|)
= (̂
∗) × (
)
(c) Take mean of generated series of data to calculate Average Marginal Effect (AME)
- Properties of ME in Probit Model
• ME also represents a probability, which is same sign as and smaller in magnitude
• ME of one explanatory variable depends on all explanatory variables
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(2) Dummy Variable
- Marginal Effect of as dummy variable
Pr( = 1|)
≈ Φ(
)
=1
−Φ(
)
=0
- Calculation of ME
(a) Calculate predicted value of index less the effect of dummy variable
̂
∗ = (
̂
) =
0
+
1
1 +
2
2 …+
−
(b) Calculate the normal distribution function, making = 1 for all observations
Φ(
0
+
1
1 +
2
2…+
) = Φ(̂
∗ +
)
(c) Calculate the normal distribution function, making = 0 for all observations
Φ(
0
+
1
1 +
2
2…+
) = Φ(̂
∗)
(d) Calculate the estimated marginal effect
Pr( = 1|)
≈ Φ(̂
∗ +
) − Φ(̂
∗)
(e) Take mean of generated series of data to calculate Average Marginal Effect (AME)
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PASS 不是我们的代言 H1 才是我们的目标
Basic Econometrics
Review Class
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Topic 1 The Basic Linear Model
1. Assumptions in Linear Regression
(1) Assumptions about the model
- MR1: The correct model is:
= 0 + 11 + 22 +⋯+ +
(2) Assumptions about the error term:
- MR2: Zero Conditional Expectation: The conditional expected value of the random errors is
zero:
(|) = 0
- MR3: Homoskedasticity: The variance of the random errors is constant and independent of
:
(|) =
2
- MR4: No Autocorrelation: Any pair of random errors are uncorrelated:
(, |, ) = 0 , = 1,2, …, ≠
- MR0: Not a necessary assumption: Normally Distributed Error Term
| ~ (0,
2)
• If you assume the error term is normally distributed, then you can write the model as:
= 0 + 11 + 22 +⋯+ + ℎ | ~ (0,
2), or
| ~ (0 + 11 + 22 +⋯+ ,
2)
(3) Assumptions about the Explanatory Variables: = (1, 2, … , )
- MR5a: The explanatory variables satisfy any one of the following:
• Non-stochastic Regressor: The explanatory variables are not random
• Exogeneity: The explanatory variables are random and uncorrelated with the error term
(, ) = 0 , = 1,2, …
- MR5b: No Exact Collinearity / No Perfect Multicollinearity: Any one of the X’s is not an exact
linear function of any of the other X’s.
2. The Gauss-Markov Theorem
- Under the assumptions of the linear regression model (MR1 – MR5), the OLS estimators ̂ =
(̂0, ̂1, ̂2,… , ̂) are the Best Linear Unbiased Estimators (BLUE) (have the smallest
variance) of all linear and unbiased estimators of = (0, 1, 2,… , )
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Topic 2 Statistical Properties & Inferences in Linear Model
Unit 1 Sampling Distribution of OLS Estimators
1. Sampling Distribution of OLS estimators
- If all MR1 – MR5 satisfied, the sampling distribution of OLS estimators:
~(,)
(
0
1
2
⋮
)
~
(
(
0
1
2
⋮
)
,
(
(0)
(0, 1)
(0, 2)
⋮
(0, )
(0, 1)
(1)
(1, 2)
⋮
(1, )
(0, 2)
(1, 2)
(2)
⋮
(2, )
⋯
⋯
⋯
⋱
⋯
(0, )
(1, )
(2, )
⋮
() )
)
2. Mean of OLS Estimators & Unbiasedness
- The mean of OLS estimators:
(|) = for = 1,2, …
if (|) = 0 for = 1,2, … (MAR2 holds)
- OLS estimators are unbiased estimators for population parameters, since (|) is equal
to , which is the parameter that is going to estimate
3. Variance of OLS Estimators
(1) Variance-Covariance Matrix
- The variance-covariance matrix of OLS estimators will be correct if all the following holds
• (|) = 0 for = 1,2, … (MAR2 holds)
• (|) =
2 (MR3 holds)
• (, |, ) = 0 for all , = 1,2, …, ≠ (MR4 holds)
• The
′ are non-random (MR5a holds)
- If all the conditions above hold, the formula for variance of OLS estimators:
() =
2
∑( − ̅)
2 =
2
( − 1)()
(2) Factors Affecting the Variance of the OLS Estimators
(a) Variation of Residuals (Positively Affected): 2
(b) Variation in Explanatory Variables (Negatively Affected): ∑( − ̅)
2
(c) Sample Size (Negatively Affected):
(d) Correlation between Explanatory Variables (Positively Affected): (, )
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Unit 2 Measuring Goodness of Fit in Linear Model
1. ANOVA Table in Regression & Estimator of Error Variance 2
Sum Square d.f. Mean Square
F Statistic for Overall
Significance
Model =∑(̂ − ̅)
2
=1
= /
=
/
/( − − 1)
Residual =∑( − ̂)
2
=1
− − 1
= /( − − 1)
Total =∑( − ̅)
2
=1
− 1
- The estimators of error variance:
̂2 =
∑ ̂
2
=1
( − − 1)
=
∑ ( − ̂)
2
=1
( − − 1)
=
( − − 1)
=
where
• : number of explanatory variables / + 1: number of parameters
• OLS residuals ̂ = ( − ̂) = − (0 + 11 + 22 +⋯+ )
2. Measuring Goodness of Fit in Linear Model
(1) Coefficient of Determination: 2
2 = 1 −
∑ ̂
2
=1
∑ ( − ̅)2
=1
= 1 −
= (, ̂)
- Coefficients of determination: 2 is the percentage of the variation in the dependent variable
about its mean that is explained by the regression model
- Problem of 2: 2 cannot decrease when including irrelevant variables in the model
(2) Adjusted 2: ̅2
̅2 = 1 −
×
− 1
− − 1
= 1 − [(1 − 2) ×
− 1
− − 1
]
- Interpretation of ̅2:
• ̅2 does not measure the percentage of the variation in the dependent variable explained
by the regression model
• ̅2 can be negative
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Unit 3: Hypothesis Testing
1. OLS t Test
- Test statistic:
=
−
()
~( − − 1)
where:
• + 1: number of explanatory variables in the model
• : value of stated in 0
• : OLS estimator for
• (): Standard error of , which is the square root of ()
- Rejection Rule:
0
Critical Value Method
Reject 0 if
p-value Method
Reject 0 if <
0: =
: ≠
< −(1 −
2
, − − 1)
> (1 −
2
, − − 1
= 2 × Pr( > ) for > 0
= 2 × Pr( < ) for < 0
: < < (, − − 1) = Pr( < )
: > > (1 − , − − 1) = Pr( > )
2. Type I Error and Type II Error
Reject 0 Not Reject 0
When 0 is True
Type I Error
Pr( 0| 0 ) =
Significance Level / Size of Test
Correct Decision
Pr( 0| 0 ) = 1 −
Confidence Level
When 0 is False
Correct Decision
Pr( 0| 0 ) = 1 −
Power of Test
Type II Error
Pr( 0| 0 ) =
- Type I error is the significance level we set
- Type II error is not under our control
• Probability of Type II error is unknown because the true parameter value is unknown
• Probability of Type II error can be reduced by larger sample size
- Probability of Type I error & Type II error is negatively related
• Probability of Type II error can be reduced by setting a larger
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3. The General OLS F Test
- The F Test Statistic
=
( − )/
/( − − 1)
~(, − − 1)
where:
• : Residual Sum of Square in restricted model
• : Residual Sum of Square in unrestricted model
• : Number of restrictions
• + 1: Number of parameters
- The Rejection Region
• Reject 0 if > (1 − ,, − − 1), or
• Reject 0 if = Pr( > ) <
4. F Test of Significance of Model
- The F Test Statistic
=
( − )/
/( − − 1)
=
( − )/
/( − − 1)
=
(2)/
(1 − 2)/( − − 1)
~( = , − − 1)
where:
• : Residual Sum of Square in restricted model
• / : Residual Sum of Square in unrestricted model
• : Total Sum of Square in unrestricted model
• : Number of restrictions
• + 1: Number of parameters
• 2: Coefficient of Determination of unrestricted model
- The Rejection Region
• Reject 0 if > (1 − , = , − − 1)
• Reject 0 if = Pr( > ) <
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Topic 3 Model Specification
➢ Review MR1: Model Specification: The correct model is:
= 0 + 11 + 22 +⋯+ +
• The functional form selection is correct
• No omission of relevant explanatory variables and
• No inclusion of irrelevant explanatory variables
Unit 1 Functional Form Selection
- Four Main Types of Functional Form:
Functional Form
Dependent
Variable
Explanatory
Variable
Interpretation of Slop
Coefficient
Linear-Linear Model
= 0 + 1 +
Unit form Unit form
One unit increase in is
associated with 1 unit increase
in
Log-Linear Model
log = 0 + 1 +
Logarithmic form Unit form
One unit increase in is
associated with 1 × 100%
percentage increase in
Linear-Log Model
= 0 + 1 log +
Unit form Logarithmic form
100% percentage increase in
is associated with 1 unit
increase in
Log-Log Model
log = 0 + 1 log +
Logarithmic form Logarithmic form
1% percentage increase in is
associated with 1% percentage
increase in (Elasticity)
- Additional Types of Functional Forms
(1) Polynomial
= 0 + 1 + 2
2 + 3
3 +
= 0 + 11 + 22 + 312 + 41
2 + 52
2 +
(2) Reciprocal
= 0 + 1
1
+
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Unit 2 Omission of Relevant Explanatory Variables
- Consider the correct model is:
= 0 + 1 + 2 +
and the incorrect model, which omit , is:
= 0 + 1 + with = 2 +
- The OLS estimation of the model above, which omit :
(1) = 1 + 2
(, )
()
• What we want to estimate is 1
• The actual estimate is: 1 + 2
(,)
()
• The difference: 2
(,)
()
is the biased amount
Unit 3 Inclusion of Irrelevant Explanatory Variables
- Consider the correct model is:
= 0 + 1 +
and the incorrect model, which included an irrelevant , is:
= 0 + 1 + 2 +
- Inclusion of irrelevance explanatory variable will cause:
• The OLS estimators of the parameters will be unbiased, because is an irrelevant
explanatory variable, which implies 2 = 0
• The variances of the OLS estimators in the model will be larger
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Unit 4 Testing for Model Specification
1. RESET Test for Model Specification
- Consider the following model with fitted model:
= 0 + 11 + 22 +
̂ = 0 + 11 + 22
- The RESET test model:
• Test reasonableness to include binomial terms:
= 0 + 11 + 22 + 1̂
2 +
with Hypothesis: 0: 1 = 0 against : 1 ≠ 0
• Test reasonableness to include binomial terms:
= 0 + 11 + 22 + 1̂
2 + 2̂
3 +
with Hypothesis: 0: 1 = 2 = 0 against : 1 ≠ 0 2 ≠ 0
- Conclusion:
• Reject 0 implies that original model specification is not correct
• Not reject 0 implies that original model specification is correct
- Including powers of ̂
2 can test:
• Incorrect functional form: Consider the reasonableness to include polynomial terms
• Omitted variable: Including polynomial terms may reflect omitted variables related to
included explanatory variables
• Note: RESET test cannot identify exact reason for model misspecification: functional form,
or omitted variables, or combination of two sources
2. Jarque-Bera Test for Normality of Error Term
- Null hypothesis and alternative hypothesis:
• 0: Errors are normally distributed
• : Errors are not normally distributed
- Test Statistic:
=
6
[2 +
( − 3)2
4
]~2( = 2)
where:
• : Sample skewness of OLS residuals
• : Sample kurtosis of OLS residuals
• : Sample size
- Conclusion of Jarque-Bera Test
• Reject 0: errors are not normally distributed and model specification is incorrect
• Not reject 0: errors are normally distributed and model specification is correct
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Unit 5: Collinear Economic Variables
➢ Review MR5b: No Exact Collinearity: Any one of the X’s is not an exact linear function of any
of the other X’s.
• Exact collinearity causes OLS estimation to be invalid
• Near-exact collinearity causes OLS estimation valid, but cause problems in estimation
1. Introduction to Collinearity
- Collinearity is the problem that two or more explanatory variables are highly correlated
- Consequence of collinearity
(1) OLS estimators are still unbiased, but variances and covariances of the OLS estimators
might be very large
(2) The individual coefficient t test statistic is not significance, but the model fitness is still
good, with high 2 and significant F test statistics
(3) Estimates may be sensitive to addition \ deletion of explanatory variables
- Solution to collinearity
• Exclude explanatory variables until no collinearities among explanatory variables
• However, there may be trade-off between omitted variables and collinearity
2. Test for Collinearity
(1) Pairwise Correlation
- Test the sample correlation between each pair of explanatory variables:
- For example, consider the model:
= 0 + 11 + 22 + 33 +
• The pairwise correlations: (1, 2) & (1, 3) & (2, 3)
• If there is correlation that is significantly different from 0, there should be collinearity
between suggested explanatory variables
(2) Auxiliary Regressions
- Construct regressions with one original explanatory variable to be new dependent variable,
and all the other original explanatory variables to be new explanatory variables
- For example, consider the model:
= 0 + 11 + 22 + 33 +
• The auxiliary regressions
1 = 0 + 12 + 23 +
2 = 0 + 11 + 23 +
3 = 0 + 11 + 22 +
• If the 2 is high in one or more equations above, there should be collinearity among
original explanatory variables
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Topic 5 Heteroskedasticity
Unit 1 Nature of Heteroskedasticity
1. Definition of Heteroskedasticity
- Heteroskedasticity: violation of MR3: The variance of error depends on data:
(|) =
2
• Variance of the error may depend on explanatory variables:
• Variance of the error may depend on other factors of data
- Consequences of heteroscedasticity
(1) The OLS estimator is still a linear and unbiased estimator but not BLUE
(2) The standard errors of OLS estimator are incorrect
(3) Hypothesis tests & confidence intervals will be misleading
- Solution to Heteroskedasticity:
• Use robust / White standard errors
• Use generalised least squares
Unit 2 Generalised Least Squares
- If we know the form of the heteroskedasticity, we can use generalised least squares
- In general, if we know that (|) =
2 = 2, we can transform the model
= 0 + 1 +
⇓
√
=
0
√
+
1
√
+
√
⇓
∗ = 01
∗ + 12
∗ +
∗
- New error term
∗ satisfies all the assumptions
• (
∗) = (
√
) =
1
√
() =
1
√
× 0 = 0
• (
∗) = (
√
) = (
1
√
)
2
() =
1
2 =
2
• (
∗,
∗) = (
√
,
√
) =
1
√
×
1
√
(, ) =
1
√
×
1
√
× 0 = 0
- The estimator for transformed model is called GLS Estimators, and is BLUE
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Unit 3 Feasible Generalised Least Squares
1. Feasible Generalised Least Squares (FGLS)
- FGLS can solve heteroskedasticity with unknown form
- Consider the following model:
= 0 + 11 + 22…+ +
we know is heteroskedasticity but we do not know the form
- The function form of heteroskedasticity by FGLS:
(|) =
2 × exp{0 + 11 + 22 …+ } =
2
•
′ are unknown parameters constructing the weight for GLS
• Use exponential for to ensure positive variance
• homoskedasticity when 1 = 2 = ⋯ =
2. Forms of FGLS
- Transformation of the model:
2 = 2 × exp{0 + 11 + 22…+ }
⇓
ln
2 = ln2 + 0 + 11 + 22…+
⇓
ln ̂
2 = ln 2 + 0 + 11 + 22…+ + ln ̂
2 − ln
2
⇓
ln ̂
2 = 0
∗ + 11 + 22…+ +
with
0
∗ = 0 + ln
2 and = ln ̂
2 − ln
2 = ln (
̂
2
2) = ln (
̂
)
2
but
~ ln
2(1)
(|) = −1.2707 ≠ 0
- Further Transformation
ln ̂
2 = 0
∗ + 11 + 22…+ +
⇓
ln ̂
2 = 0
∗ − 1.2707 + 11 + 22…+ + + 1.2707
⇓
= ln ̂
2 = ̃0 + 11 + 22…+ +
∗
with
̃0 = 0
∗ − 1.2707 and
∗ = + 1.2707
here
(
∗|) = 0
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3. Steps in FGLS
(1) Estimate the original model and get residuals: ̂ = ̂ −
= 0 + 11 + 22…+ +
(2) Use residuals to estimate:
= ln ̂
2 = ̃0 + 11 + 22…+ +
∗
Note:
• Estimated intercept will be off by -1.2707
• Estimated slope will be consistent (valid in large sample)
(3) Use fitted value of to run GLS:
√exp(̂)
=
0
√exp(̂)
+
11
√exp(̂)
+
22
√exp(̂)
…+
√exp(̂)
+
√exp(̂)
with
2 = (|) =
2 × exp()
(
1
√exp(̂)
) = (
1
√exp(̂)
)
2
() =
2
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Unit 4 Detecting Heteroskedasticity
1. Plot Method
(1) Plot of against with fitted line
(2) Plot of against residuals ̂
2. White’s Test
- White’s test can test heteroskedasticity with unknown form by comparing White’s standard
error & OLS standard error
- Hypothesis:
0:
2 = 2 (Homo) against :
2 ≠ 2 (Homo) against
with
•
2: White’s standard error
• 2: OLS standard error
- Test Statistics
• If the model is:
= 0 + 11 + 22 +
• The aux regression of squared residuals on all possible combinations of
′
̂
2 = 0 + 11 + 22 + 31
2 + 42
2 + 512 +
• The test statistics:
× 2~2( − 1)
with
: Sample Size
2: 2 of aux regression
: Number of parameters in aux regression
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- Problem of White’s test
• White’s test is in fact a test for model specification
• 0 assume that
• Error is homoskedasticity
• Error in independent of
′
• Functional form is correct
• Reject 0 implies that one or more conditions above not satisfied, which may or may not
identify heteroskedasticity
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Topic 6 Time Series
Unit 1 Autocorrelation
1. The Nature of Autocorrelation
- The problem of autocorrelation in error term:
• Violation of MR4: Error terms are correlated over-time (, −) ≠ 0 for some
- Consequence of autocorrelation
(4) The OLS estimator is still a linear and unbiased estimator but not BLUE
(5) The standard errors of OLS estimator are incorrect
(6) Hypothesis testing & confidence interval are invalid
- Solution to autocorrelation:
• Use HAC standard errors
• Use generalised least squares: with known form of autocorrelation
2. First-Order Autoregressive Process in Error Term: (1)
- If we know that the error term is autocorrelated one-period-ahead
(, −1) ≠ 0
we can express its form in (1) time-series regression:
= −1 +
- The error term is composed of two parts:
• −1: The carry-over from previous period
• : The correlation in
• −1 < < 1: The carry-over will diminish period by period
• : The new shock this is uncorrelated overtime, and satisfies Gauss-Markov Theorem
- Properties of First-Order Autoregressive Errors:
• Zero mean
() = 0
• Homoscedasticity
() =
2 =
2
1 − 2
• Covariance between successive error terms:
(, −) =
2 =
2
1 − 2
• Correlation between successive error terms:
(, −) =
(, −)
√()√(−)
=
2
√2√2
=
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3. Generalised Least Square
(1) Methods of Generalised Least Square
- The time series regression model:
= 0 + 1 + with = −1 +
with properties
• () = 0
• () =
2 =
2
1−2
• (, −) =
2 =
2
1−2
- Substitute the (1) error into the model:
= 0 + 1 +
= 0 + 1 + (−1 + )
−1 = −1 − 0 − 1−1⟸ −1 = 0 + 1−1 + −1
⟹ = 0 + 1 + (−1 − 0 − 1−1) +
− −1 = 0(1 − ) + 1( − −1) +
- The transformed model:
∗ = 0
∗ + 1
∗
∗ +
with
∗ = − −1; 0
∗ = 0(1 − ); 1
∗ = 1;
∗ = − −1
where
• in the new model satisfies Gauss-Markov Theorem
• (T-1) observations are included in new model
(2) Steps in Generalised Least Square
➢ Step 1: Estimate by OLS
- Since = −1 + , we ca estimate using OLS residuals as estimation of error term
• Obtain the OLS residuals
̂ = − 0 − 1
• Put OLS residuals into (1) model
̂ = ̂−1 +
• Estimation of
̂ =
∑ ̂̂−1
=2
∑ ̂−1
2
=2
➢ Step 2: Implement GLS
- The transformed model:
∗ = 0
∗ + 1
∗
∗ +
with
•
∗ = − −1;
• 0
∗ = 0(1 − );
•
∗ = − −1
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4. Test for Autocorrelation – LM Test
- Consider the following time series model:
= 0 + 1 +
where we want to test whether: (, −) ≠ 0 for some ≥ 1
- LM Test for (1) errors: = −1 +
• Hypothesis: 0: = 0 against 1: ≠ 0
• Test Statistics: Using 2 of the following model
̂ = 0 + 1 + ̂−1 +
= × 2~2(1)
• Reject 0 if >
2(, 1)
- LM Test for (2) errors: = 1−1 + 2−2 +
• Hypothesis: 0: 1 = 2 = 0 against 1: At least one ≠ 0, = 1,2
• Test Statistics: Using 2 of the following model
̂ = 0 + 1 + 1̂−1 + 2̂−2 +
= × 2~2(2)
• Reject 0 if >
2(, 2)
- Test for () errors: = 1−1 + 2−2 +⋯+ − +
• Hypothesis: 0: 1 = 2 = ⋯ = = 0 against 1: At least one ≠ 0, = 1,2…
• Test Statistics: Using 2 of the following model
̂ = 0 + 1 + 1̂−1 + 2̂−2 +⋯+ ̂− +
= × 2~2()
• Reject 0 if >
2(, )
- Problem of LM Test
• LM tests need us to know the form of autocorrelation
• If we do not know the form of autocorrelation:
• Testing Down Approach
• Testing up a pp roach
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Unit 2 Non-Stationary Time Series
- In time series models, all time series data included should be stationary
- Summary: Time Series Patterns
Name Plot Stationarity Unit Root ADF Test Type
Stationary
Time
Series
/
Mean-
Reverting
Stationary No Drift
Trend
Stationary
Non-Stationary No Drift
Random
Walk
Non-Stationary Yes Trend
Random
Walk with
Drift
Non-Stationary Yes Trend
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1. Stationary Time Series
(1) Definition of Stationary
- A time series data is stationary if:
• Its mean is constant overtime
() =
• Its variance is constant overtime
() =
2
• Its autocovariance for any given lag length is constant overtime
(, −) = (, +)
(2) Stationary Time Series in (1) Model
- The (1) model of which is stationary:
= −1 + with || < 1
where satisfies Gauss-Markov Theory
- Then, we have:
• () = 0
• () =
2 (
1
1−2
)
• (, +) =
2 (
1−2
)
(3) Different Kinds of (1) Model with Stationary Time Series
- (1) Model with zero mean
= −1 +
- (1) Model with non-zero mean
= + −1 + with () = =
1−
• De-meaning:
= + −1 +
= (1 − ) + −1 +
− = (−1 − ) +
with = (1 − )
- (1) Model with non-zero mean & linear trend (Trend Stationary which is Non-Stationary)
= + + −1 + with () = +
• De-meaning & de-treading:
= + + −1 +
= [(1 − ) + ] + (1 − )
− − = [−1 − − ( − 1)] +
with = (1 − ) + and = (1 − )
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2. Non-Stationary Time Series
- In the (1) model of , if = 1, then is non-stationary
(1) Random Walk Model
= −1 +
• Apply recursive substitution
= 0 +∑
−1
=0
then
() = (0) = 0
() = (0 +∑
−1
=0
) = 2
• Random walk is non-stationary because its variance increases over time
(2) Random Walk with Drift
= + −1 +
• Apply recursive substitution
= + 0 +∑
−1
=0
then
() = + 0
() =
2
• Random walk is non-stationary because its mean and variance increases over time
3. Spurious Regression
- Definition of Spurious Regression
• If we regress two non-stationary time series on each other, we will get a significant OLS
coefficient even if the series are unrelated
• Consider the following two random walk time series
= −1 + 1
= −1 + 2
and are uncorrelated, however in the following model:
= 0 + 1 +
estimate of 1 is usually statistically significant suggesting and are correlated
- Consequence of Spurious Regression
• OLS estimators’ statistical properties are not valid
• OLS estimators’ statistical inferences (hypothesis testing) are not correct
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4. Testing for Non-Stationarity: Dickey Fuller (DF) Test
(1) Dickey Fuller (DF) Test
- Consider the model:
= −1 +
or
Δ = ( − 1)−1 +
= −1 +
- Hypothesis:
0: Non-Stationary / = 1 / = 0
0: Stationary / < 1 / < 0
- Different Kinds of Models
• DF Test I: If time series appears to fluctuate around mean zero:
Δ = −1 + with 0: = 0
• DF Test II: If time series appears to fluctuate around non-zero mean:
Δ = + −1 + with 0: = 0
• DF Test III: If time series appears to fluctuate around linear trend:
Δ = + + −1 + with 0: = 0
(2) Augmented Dickey Fuller (ADF) Test
- ADF test consider possible autocorrelation in error term
- We add augmented terms (lagged first difference terms in ) to eliminate autocorrelation
Δ = + −1 + ∑ Δ−
=1 + with 0: = 0
(3) Test Statistics
- The test statistics for DF test & ADF test follows DF distribution, not t distribution
- We just focus on p-value of the tests
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Topic 7 Stochastic Regressors
Unit 1: Stochastic Regressors and Problem of Endogeneity
1. Definition of Endogeneity
- MR5a version 2: Exogeneity requires that the explanatory variables are random and
uncorrelated with the error term:
( , ) = 0
- Endogeneity is a problem that explanatory variable is correlated with error term:
( , ) ≠ 0
2. Consequence of Endogeneity
- OLS estimators are biased and inconsistent
- OLS standard errors are incorrect
- Hypothesis tests & confidence interval are invalid
3. Sources of Endogeneity
(1) Classical Measurement Error
(2) Omitted Variables
(3) Lagged Dependent Variable with (1) Error Structure
(4) Simultaneous Causal Equations
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Unit 2: Sources of Endogeneity
1. Classical Measurement Error
- Classical measurement error arises when explanatory variables are measured with error
- Consequences of Classical Measurement Error
• OLS estimators are biased, and standard errors are incorrect
• Attenuation Bias: 1 is biased towards zero
• No bias if and only if no measurement error
2. Omitted Variables
- Omitted variable bias arise if
• One or more relevant explanatory variables are omitted from the true mode, and
• Omitted explanatory variables are correlated with included explanatory variables
- Consider the correct model is:
= 0 + 1 + 2 +
with
• (,) = 0
• (, ) = 0
- Consider the incorrect model, which omit :
= 0 + 1 + (2 + )
= 0 + 1 +
- In this estimated model:
(, ) = (, 2 + )
= 2(,) + ( , )
= 2(,)
Endogeneity arises if 2 ≠ 0 and (,) ≠ 0
- The incorrect OLS estimation:
(1) = 1 + 2
(,)
()
The bias:
= 2
(,)
()
3. Lagged Dependent Variables
- In time series model, when including a lag of dependent variable as an explanatory variable:
= 0 + 1−1 +
- If there is autocorrelation in error term
• correlated with −1
• −1 correlated with −1: endogeneity
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Unit 3: Instrumental Variable (IV) and Two Stage Least Squares (2SLS)
- Instrumental Variable (IV) and Two Stage Least Squares (2SLS) is the effective and efficient
method to solve all sources of endogeneity
1. Instrumental Variable Validity Condition
- Consider the model where is endogenous
= 0 + 1 +
- Consider the instrumental variable , that satisfies IV validity conditions
(a) Exclusive Restriction
• does not directly correlated with dependent variable
• does not correlated with error term: (, ) = 0
(b) Instrumental Relevance
• is correlated with endogenous variable ( , ) ≠ 0
2. Two Stage Least Square
- Consider the model where is endogenous and is a valid IV for
= 0 + 1 +
- The process of Two Stage Least Squares (2SLS) estimation:
(a) Stage 1: Regress on to get a fitted value of :
̂ = ̂0 + ̂1
(b) Stage 2: Estimate original model, replacing with fitted value of
̂ = 0 + 1̂
- By the process of 2SLS, the estimator 1 is called 2SLS estimator or IV estimator
3. Properties of IV Estimators
(1) IV estimators are consistent if IV validity conditions satisfied
(2) In small sample, IV estimators are biased
(3) In large sample, IV estimators are approximately unbiased and normally distributed
- In the model:
= 0 + 1 +
where is endogenous and is a valid IV for
- The sampling distribution of IV estimator in large sample
1
. ~ (1,
2
2
2 )
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(4) The Variance of IV Estimator
- In large sample, IV estimators have variance:
(1
) =
2
2
2
where
•
2: Population variance of
•
2: Population variance of
•
2 : Square of population correlation between and
- Estimates of Variance of IV Estimator
̂(1
) =
̂
2
2
2 =
̂
2
∑( − ̅)2
2
where
• ̂
2 : Sample variance of residuals
•
2: Sample variance of
•
2 : Square of sample correlation between and
- Properties of Variance of IV Estimator
(a) Suppose (, ) = 0, then for the OLS estimator & IV estimator
̂(1
) =
̂
2
∑( − ̅)2
=
2 ̂(1
)
When OLS is valid, OLS estimator will have a lower variance than IV estimator
(b) Higher sample correlation
2 lowers the estimated variance of IV estimator
(c) Larger sample size lower variance of IV estimator
(d) Greater variance in lower variance of IV estimator
4. Weak Instruments
- Weak Instrument: (, ) ≈ 0
- How to detect weak instrument: Rule of thumb: F test statistic of IV in stage 1 less than 10
- Problem of weak instrument:
• Normal distribution provides poor approximation to sampling distribution of the IV
estimator in large sample
• No justification for using t-tests and F-tests which are based on assumption of normality
• Statistical inference: affects conclusions from hypothesis tests, which are based upon an
incorrect distribution of the test statistic when 0 is true
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Unit 4: Simultaneous Causal Equation System
1. Definition of Simultaneous Causal Equations
- Simultaneous Causal Equations: Two variables are jointly determined
- Simultaneous causal equations is one source of endogeneity: (, )
2. Simultaneous Causal Equations: Structural Equations
- Consider two simultaneous caused variables: 1 & 2
{
1 = 12 + 11 + 1 (1)
2 = 21 + 22 + 2 (2)
(1) Endogenous Variables
- In the equations system above, endogenous variables:
• 2 is the endogenous variable in equation 1
• 1 is the endogenous variable in equation 2
• 1 & 2 are both the endogenous variables in the system
(2) Exogenous Variables & Instrumental Variables
- In simultaneous causal equation system, all exogenous variables in the system can be
considered as instrumental variables, irrespective of which equation they are in
- In the equations system above, exogenous variables:
• 1 & 2 are the exogenous variables / instrumental variables in the system
• 1 is the exogenous variable in only equation 1
• 1 is used to identify equation 1
• 1 can be used as exclusive IV for 1
• 2 is the exogenous variable in only equation 2
• 2 is used to identify equation 2
• 2 can be used as exclusive IV for 2
(3) Identification
- In simultaneous equation system, an equation is identified if it contains exogenous variable
that does not exist in other equations
- In a system of 2 simultaneous equations with 2 endogenous variables, at least 1 equations
must be identified; we need at least 1 exogenous variables
- If a simultaneous equation system is identified, the parameters in this system can be
estimated consistently
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3. Simultaneous Causal Equations: Reduced Forms
(1) Definition and Illustration of Reduced Forms
- The reduced forms express each endogenous variable in terms of all exogenous variables in
the system
- Consider the structural equations:
{
1 = 12 + 11 + 1 (1)
2 = 21 + 22 + 2 (2)
assuming:
• (1) = (2) = 0
• (1, 2) = 0
- Reduced forms:
{
1 = 111 + 122 + 1 (1)
2 = 211 + 222 + 2 (2)
where coefficients ′ are functions of structural coefficients
(2) Properties of Reduced Form
- Reduced form can be estimated by OLS, because no endogeneity in reduced form
- Reduced form is actually the stage 1 of 2SLS
- Reduced form parameters are non-linear function of structural parameters
• We cannot estimate structural parameters
• We can estimate non-linear function of structural parameters
4. Two-Stage Least Square
(1) 2SLS in Simultaneous System with Two Equations
- Consider the structural equations
{
1 = 12 + 11 + 1 (1)
2 = 21 + 22 + 2 (2)
(a) Estimation of Equation 1
- Endogenous Variable: 2
- Instrumental Variable of 2: 2, with IV validity conditions
• Exclusion Restriction: (2, 1) = 0: 2 won’t directly correlated with 1
• Instrumental Relevance: (2 , 2) ≠ 0: 2 correlated with 2
- Two Stage Least Square
• Stage 1: Regress 2 on all IVs in the system and obtain fitted value
̂2 = ̂211 + ̂222
• Stage 2: Estimate equation (1), replacing 2 with fitted value
̂1 = ̂1̂2 + ̂11
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(b) Estimation of Equation 2
- Endogenous Variable: 1
- Instrumental Variable of 1: 1, with IV validity conditions
• Exclusion Restriction: (1, 2) = 0: 1 won’t directly correlated with 2
• Instrumental Relevance: (1, 1) ≠ 0: 1 correlated with 1
- Two Stage Least Square
• Stage 1: Regress 1 on all IVs in the system and obtain fitted value
̂1 = ̂111 + ̂122
• Stage 2: Estimate equation (2), replacing 1 with fitted value
̂2 = ̂2̂1 + ̂22
(2) Properties of Two Stage Least Square
• The 2SLS estimator is biased but consistent
• In large samples, 2SLS estimator is approximately normally distributed
• The variances and covariances of the 2SLS estimator are unknown in small samples but
can be approximated in large sample
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Topic 8 Panel Data Model
Unit 1: Panel Data Concepts
1. Introduction to of Panel Data
- Panel data is the combination of cross-sectional data & time-series data
• N: Number of cross-sectional units
• T: Number of time periods
2. Variables in Panel Data Frame
- : Time-variant & Individual-variant
- : Time-invariant & Individual-variant
- : Time-variant & Individual-invariant
- : Time-invariant & Individual-invariant
3. Estimation of Panel Data Model
- Consider a general panel data regression:
= 0 + 11 + 22…+ +
The purpose is to estimate that reveals relationship between & that is applicable
across all individuals & across all time periods
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Unit 2: Pooled OLS Model
1. Definition of Pooled OLS
- Pooled model: Simple OLS applied on whole panel data set
- Pooled OLS model example:
= 0 + 11 + 22 +
where
• : Individual Index: = 1,2, …
• : Time Index: = 1,2, …
• Total observations: ×
2. Assumptions of Pooled OLS
- Required assumptions for pooled OLS estimator to be consistent
- Assumptions on individual dimension:
• (|) = 0 Zero Mean
• (|) =
2 Homoskedasticity across individuals
• (, ) = 0 Exogeneity
• (, ) = 0 No autocorrelation across individuals
- Assumptions on time dimension:
• (, |) = ≠ 0 Autocorrelation across time
• (, |) = () = Heteroskedasticity across time
3. Problem of Heteroskedasticity & Autocorrelation across Time
- Consequence of Pooled OLS
• The OLS estimator is still unbiased & consistent but not BLUE
• The standard errors are incorrect
• The hypothesis testing & confidence interval based on these standard errors are invalid
- Solution: use cluster-robust standard errors
• Clusters: the time series observations on individuals
• Cluster-robust standard errors are only valid in large sample
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Unit 3: Fixed Effects Model
1. Fixed Effects Model
- Fixed effect model treats individual time-invariant characteristics as a constant parameter
- Fixed effects model:
= 0 + 11 + 22…+ +
2. Fixed Effect Estimation: Dummy Variable Method
- Fixed effects model by individual dummy variables:
= 0 + 11 + 22…+ +
where
= {
1 for individual j
0 othervise
- Required assumptions for OLS estimator to be consistent
• (|) = 0 Zero Mean
• (|) =
2 Homoskedasticity
• (, |) = 0 No autocorrelation
• (, ) = 0 Exogeneity
- As sample size increase:
• Estimation of 1, 2, … , are consistent
• Estimation of 0
′ are inconsistent
3. Fixed Effect Estimation: Within-Transformation
- Within-transformation method keeps only slope coefficients and remove individual intercepts
- Process of within-transformation
• Fixed effect model:
= 0 + 11 + 22…+ + (1)
• Take time average on each term:
1
∑
=1
=
1
∑0
=1
+
1
∑11
=1
+
1
∑22
=1
…+
1
∑
=1
+
1
∑
=1
̅ = 0 + 1̅1 + 2̅2…+ ̅ + ̅ (2)
• Take (1) – (2):
− ̅ = 1(1 − ̅1) + 2(2 − ̅2) + ⋯+ ( − ̅) + ( − ̅)
∗ = 11
∗ + 22
∗ …+
∗ +
∗
- The final version is the within-transformed model:
∗ = 11
∗ + 22
∗ …+
∗ +
∗
- Estimation of within-transformed model gives within estimator
• Individual intercepts are removed
• Time-invariant variables will also be removed
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Unit 4: Random Effect Model
1. Definition of Random Effect Model
- Random effect model treats individual time-invariant characteristics as a random variable
- Random effect model
= + 11 + 22…+ +
- Assumptions about the model:
• (|, ) = 0 Zero Mean
• (|, ) =
2 Homoskedasticity
• (, |, ) = 0 ≠ No autocorrelation across time
• (, |, ) = 0 ≠ ≠ No autocorrelation across individual
• (, |) = 0 Exogeneity
- Assumptions about the individual random effect :
• (, ) = 0: & are independent
• (|) = 0: Expectation of is the intercept
• (|) =
2: Constant Variance
• ( , ) = 0 ≠ : No correlation among individuals
• (, ) = 0: Uncorrelated with error term
2. Random Effect Model Estimation: Generalised Least Squares
(1) Transform the Random Effect Model
- Consider the following random effect model:
= + 11 + 22 +
- Add & Subtract 0:
= 0 − 0 + + 11 + 22 +
= 0 + 11 + 22 + ( − 0 + )
= 0 + 11 + 22 +
- In the final version:
= 0 + 11 + 22 +
• Mean & Variance of Error Term:
(|, ) = (|) − 0 + (|, ) = 0 − 0 = 0
(|, ) = () + () + 2(, ) =
2 +
2
• Autocovariance of Error Term:
(, ) = 0: No autocorrelation across individual
(, ) =
2: Autocorrelation across time
(, ) = 0: No autocorrelation across individual & across time
• Autocorrelation of Error Term across Time:
= (, ) =
(, )
√()()
=
2
2 + 2
Interpretation: proportion of total variance in the error term that can be attributed
to variance in the individual time-invariant components
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(2) GLS Estimation of Random Effects Model
- Transformed random effects model:
= 0 + 11 + 22 +
Define:
= 1 −
2
√2 + 2
Then estimate transformed model to obtain estimation of ̂
2 & ̂
2 and calculate ̂
- The GLS mode to be estimated:
− ̂̅ = (1 − ̂)0 + 1(1 − ̂̅1) + 2(2 − ̂̅2) + ( − ̂̅)
∗ = 0
∗ + 11
∗ + 22
∗ +
∗
- Properties of Random Effects Estimators (GLS)
• Uses both within individual and between individual variation to identify coefficients
• Could estimate time-invariant variables
• GLS estimator generally provides estimates with lower standard errors
3. Correlated Random Effects (CRE)
- Consistency of Random Effects Estimators:
• The random effects estimator will be consistent if the random error = +
uncorrelated with any of the explanatory variables
• If this is not satisfied, then the random effects estimator is biased and inconsistent
- Consider the following original random effect model
= 11 + 22 + +
with correlated with explanatory variables:
= + 1̅1 + 2̅2 +
then substitute in the original model
= + 11 + 22 + 1̅1 + 2̅2 + +
then estimate this model by random effect estimator
- The CRE estimators of 1 & 2:
• Will be identical to fixed effect estimator if 1 & 2 is not zero
• Will be identical to random effect estimator if 1 & 2 is zero
• We can test this by 0: 1 = 2 = 0
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Unit 5: Comparisons among Panel Data Models
1. Comparison of Models:
Pooled OLS Fixed Effects Random Effects
Time-invariant
Unobserved
Heterogeneity
Not Allow Allow
Treat as constant coefficient
Correlated with
Allow
Treat as random variable
Uncorrelated with
Autocorrelation across
Time
May exit
Use cluster-robust SE
May exit
Use cluster-robust SE
Must exit
Generalised Leased
Squares
Identification of
Variation
Within individual variation Within individual
variation & Between
individual variation
Variables Included Not allow Allow
2. Choose between Models
(1) Pooled OLS Model vs Fixed Effect Model
- Fixed effects model by individual dummy variables:
= 0 + 11 + 22…+ +
- Hypothesis:
• 0: Fixed effects 0
′ are jointly zero
• 1: At least one fixed effect 0
′ are not zero
- Test Statistic: F Test
- Conclusion:
• Reject 0: Fixed effects is the appropriate model
• Not Reject 0: Pooled OLS is the appropriate model
(2) Pooled OLS Model vs Random Effect Model
- The variance of random effect: () =
2
- Hypothesis:
• 0:
2 = 0
• 1:
2 ≠ 0
- Test Statistic: 2(1)
- Conclusion:
• Reject 0: Random effects is the appropriate model
• Not Reject 0: Pooled OLS is the appropriate model
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(3) Fixed Effect Model vs Random Effect Model
- FE Estimator & RE Estimator
• When ( , ) = 0: Both FE & RE consistent but RE has lower variance
• When ( , ) ≠ 0: Only FE consistent
- Hausman Test: Compare difference between FE Estimator & RE Estimator
- Hypothesis or one pair of parameters:
• 0: , = ,
• 1: , ≠ ,
- Test Statistic for each pair of parameters:
=
, − ,
√(, − ,)
Combined test statistic asymptotically follows 2()
- Conclusion:
• Reject 0: Fixed effects is the appropriate model
• Not Reject 0: Random effects is the appropriate model
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Topic 9 Count Data Model
1. Model for Count Variable:
- Dependent variable is a count variable that take non-negative integer values: 0,1,2,…
- Characteristics of Count Data:
• Outcome has no natural upper bound
• Outcome will likely be zero for a non-trivial proportion of the sample
• Outcome is often right skewed
2. The Poisson Regression Model
- Poisson distribution can explain and predict the probability of count variable
- The Poisson parameter:
= exp(0 + 11 + 22…+ )
- The dependent variable follows Poisson distribution on parameter :
| ~ ()
with probability mass function
Pr( = |) = ( = |) =
− ×
!
- Expected Value & Variance
• (|) =
• (|) = This is the intrinsic heteroskedasticity: solve by robust standard error
3. Estimation of Poisson Model
- Use maximum likelihood estimation (MLE), rather than OLS, to estimate Poisson Model
- The Likelihood Function: represents the probability of observing the data:
(0, 1, … ) = (1, 2, … ) = (1)(2)…()
=
−1 × 1
1
1!
×
−2 × 2
2
2!
…×
− ×
!
- The MLE choose estimators of (0, 1, …) that can maximize the Likelihood Function
- Properties of MLE estimator in large sample:
• Normally distributed
• Consistent
• Best: has the smallest variance
4. Prediction of Poisson Model
- Prediction of Conditional Mean:
(|)̂ = ̂ = exp(̂0 + ̂11 + ̂22…+ ̂)
- Prediction of Probability
Pr( = |)̂ =
−̂ × ̂
!
ℎ = 0,1,2, …
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5. Marginal Effects in Poisson Model
(1) Continuous Variable:
- Marginal Effects of : Semi-Elasticity
%∆(|)
∆
= 100 × [
1
(|)
×
(|)
] = 100 ×
Interpretation: One unit increase in is expected to cause × 100% percentage
increase in average number of
- Marginal Effects of ln : Elasticity
%∆(|)
∆ ln
≈
%∆(|)
%∆
=
Interpretation: One percent increase in is expected to cause % percentage increase
in average number of
(2) Dummy Variable
- Marginal Effect of as Dummy Variable
%∆(|)
∆
≈ 100 × [
1
(|)
×
(|)
] = 100 × ( − 1)
Interpretation: Moving from = 0 to = 1 is expected to cause (
− 1) × 100%
percentage increase in average number of
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Topic 10 Binary Outcome Model
Unit 1: Introduction to Binary Outcome Model
- Dependent variable is a Bernoulli variable (dummy variable) that take either 0 or 1
- Distribution of :
= {
1 with probability
0 with probability 1 −
with pmf
() =
× (1 − )
(1−) = {
ℎ = 1
1 − ℎ = 0
mean & expectation
• () =
• () = (1 − ) Intrinsic Heteroskedasticity
- The fitted value ̂: probability
Unit 2: Linear Probability Model (LPM)
1. Introduction to Linear Probability Model
- Regression Model:
= 0 + 11 + 22…+ +
- Conditional Mean Function:
(|) = = 0 + 11 + 22…+
- Random Error: Bernoulli Distributed: Not Normally Distributed
= − (|) = {
1 − (0 + 11 + 22…+ ) ℎ = 1 ℎ
−(0 + 11 + 22 …+ ) ℎ = 0 ℎ 1 −
2. Heteroskedasticity in LPM
(1) Heteroskedasticity
- The variance of error term in LPM:
(|) = (1 − ) =
2
- The form of heteroskedasticity is in a quadratic manner with
- Consequence of heteroskedasticity:
• Standard errors for OLS estimator are incorrect
• Confidence interval & hypothesis testing are misleading
- Solution:
• Huber-White / Robust Standard Error
• Feasible Generalised Leased Square
(2) Feasible Generalised Leased Square
- Steps of FGLS for LPM:
• Estimate LPM by OLS: = 0 + 11 + 22…+ +
• Calculate ̂ = ̂0 + ̂11 + ̂22…+ ̂
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• Estimate the weighted model:
√(1 − )
=
0
√(1 − )
+
11
√(1 − )
+
22
√(1 − )
…+
√(1 − )
+
√(1 − )
- Problem of FGLS
• When ̂ = 0 or ̂ = 1: weighting series is undefined
• When ̂ outside range [0,1]: weighting series is undefined
3. Marginal Effect of LPM
- Marginal Effect
=
(|)
=
Pr( = 1|)
- Interpretation:
• Continuous Variable: One unit increase in cause increase in probability for = 1
• Dummy Variable: Moving from = 0 to 1 cause increase in probability for = 1
• Intercept: Average probability for = 1 is
4. 2 of LPM
- 2 in LPM is high if observations are clustered around mean of (low variance of )
- Trade-off:
• High 2 ⟹ Low variance of ⟹ Low precise of estimate
• Low 2 ⟹ High variance of ⟹ High precise of estimate
5. Problem of LPM
- Predicted value ̂ may outside range [0,1] and lead to a negative predicted variance
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Unit 3: Probit Model
1. Introduction to Probit Model
- Probit Model: Fit data in cumulative distribution function of standard normal distribution
Pr( ≤ 0 + 11 + 22…+ )
= Φ(0 + 11 + 22 …+ )
- Slope of Φ(0 + 11 + 22…+ ) is not constant, which increase rapidly then slowly
2. Latent Variable Foundations for Probit Model
- Observed Dependent Variable
= {
1 ′′
0 ′′
- Latent Variable:
∗: Linear Model:
∗ = 0 + 11 + 22…+ + with | ~ (0,
2)
•
∗ ∈ (−∞,+∞) which can take any real value
•
∗ is unobserved
- Relationship between Observed Dependent Variable & Latent Variable:
= {
1
∗ > 0
0
∗ ≤ 0
Pr( = 1) = Pr(
∗ > 0) = Φ(
0 + 11 + 22…+
)
Pr( = 0) = Pr(
∗ ≤ 0) = 1 − Φ(
0 + 11 + 22…+
)
• Index Function: 0 + 11 + 22…+
• Index function is large & positive: Φ(. ) → 1
• Index function is large & negative: Φ(. ) → 0
• Index function is zero: Φ(. ) = 0.5
• Interpretation of
∗: A possible dependent variable of which is continuous
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- Probit Model is a link between Normal Distribution & Bernoulli Distribution
• The distribution of is Bernoulli Distribution
= {
1 with probability
0 with probability 1 −
where
= Φ(
0 + 11 + 22…+
)
• The pmf of :
() =
× (1 − )
(1−) = {
ℎ = 1
1 − ℎ = 0
3. Estimation of Probit Model: Maximum Likelihood Estimation (MLE)
- Maximum Likelihood Estimation (MLE) choose (0, 1, … ) that maximize
• Likelihood Function:
(0, 1, … ) = (1, 2, … ) = (1)(2)…()
• log Likelihood Function:
ln (0, 1, … ) = ln (1, 2, … ) = ln (1) + ln (2)…+ ln ()
where
() = {
Φ(
0 + 11 + 22…+
) ℎ = 1
1 − Φ(
0 + 11 + 22…+
) ℎ = 0
- Properties of MLE estimator in large sample:
• Normally distributed
• Consistent
• Best: has the smallest variance
- Note: MLE in R
• Coefficients reported are divided by : (
̂0
), (
̂1
), …, (
̂
)
• Reports Z-statistic rather than t-statistic, because of large sample
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4. Truth Table
- After estimation, we generate predicted model
̂ = Φ(
0
+
1
1 +
2
2…+
)
- Then predicted dependent variable:
̂ = {
1 ̂ ≥
0 ̂ <
This decision may lead to correct or incorrect prediction
- Truth Table:
Number of Values
Truth
= 1 = 0
Prediction
̂ = 1
a
predicted correctly for = 1
c
predicted incorrectly for = 0
̂ = 0
b
predicted incorrectly for = 1
d
predicted correctly for = 0
• Proportion of correct prediction of = 1:
+
• Proportion of correct prediction of = 0:
+
5. Marginal Effect of Probit Model
(1) Continuous Variable
- Marginal Effect of as continuous variable
Pr( = 1)
=
Φ(
)
(
)
×
(
)
= (
0
+
1
1 +
2
2…+
) × (
)
- Calculation of ME
(a) Calculate predicted value of index
̂
∗ = (
̂
) =
0
+
1
1 +
2
2 …+
(b) Calculate estimated marginal effect
Pr( = 1|)
= (̂
∗) × (
)
(c) Take mean of generated series of data to calculate Average Marginal Effect (AME)
- Properties of ME in Probit Model
• ME also represents a probability, which is same sign as and smaller in magnitude
• ME of one explanatory variable depends on all explanatory variables
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(2) Dummy Variable
- Marginal Effect of as dummy variable
Pr( = 1|)
≈ Φ(
)
=1
−Φ(
)
=0
- Calculation of ME
(a) Calculate predicted value of index less the effect of dummy variable
̂
∗ = (
̂
) =
0
+
1
1 +
2
2 …+
−
(b) Calculate the normal distribution function, making = 1 for all observations
Φ(
0
+
1
1 +
2
2…+
) = Φ(̂
∗ +
)
(c) Calculate the normal distribution function, making = 0 for all observations
Φ(
0
+
1
1 +
2
2…+
) = Φ(̂
∗)
(d) Calculate the estimated marginal effect
Pr( = 1|)
≈ Φ(̂
∗ +
) − Φ(̂
∗)
(e) Take mean of generated series of data to calculate Average Marginal Effect (AME)
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Basic Econometrics 2022S1 期末班
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