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C29IE
Page 1 of 10
Semester 1 – 2021/22
SCHOOL OF SOCIAL SCIENCES
Edinburgh Business School
Course Name: INTRODUCTORY ECONOMETRICS
Course Code: C21IE
SEMESTER ONE – 2021/22
INSTRUCTIONS
Section A – Answer BOTH questions
Section B – Answer any FIVE questions
Section C – Answer only ONE question
C29IE
Page 2 of 10
Semester 1 – 2021/22
Section A (THEORY) (26 marks)
Q1. This question has two parts, (a) and (b). Answer ALL parts.
(a) What is a confidence interval and how should it be interpreted? Explain how we
can use our knowledge of the sampling distribution of the OLS estimator ̂ to
construct a confidence interval.
(b) What is “omitted variable bias”? What is a “control variable”? Explain how
control variables can be used to address the failure of Least Squares Assumption
#1 for Causal Inference, i.e., the failure of the assumption that E(ui|Xi)=0.
Section B: ANSWER ANY 4 OF THE FOLLOWING 9 QUESTIONS (24 marks, 6
marks per answer)
Q2: Least Squares Assumption #1 for Prediction is: “The out-of-sample observation
(XOOS,YOOS) is drawn from the same distribution as the estimation sample
(Xi,Yi), i = 1,…, n.” What are the consequences for prediction based on OLS if
this assumption fails?
Q3: What are “robust standard errors” and why do we use them?
Q4: Explain what is meant by pseudo-out-of-sample forecast.
Q5: What is the Lasso and when should we use it?
Q6: Explain what it means for a time series to be stationary.
Q7: Define AIC (Akaike Information Criterion) and discuss how it can be used to
select the number of lags in a time series regression.
Q8: What is a DAG and how can it be used for causal inference? Provide an
example of a simple DAG.
Q9: What is the method of “differences-in-differences”? Give an example of its use.
Q10: Given an example of a measure of “goodness of fit”, and explain what it means
and how it can be used.
C29IE
Page 3 of 10
Semester 1 – 2021/22
Section C: ANSWER ANY 1 OF THE FOLLOWING 2 QUESTIONS (50 marks)
Q11: Prediction and causal inference
Q12: Time series
(Section C questions follow below.)
C29IE
Page 4 of 10
Semester 1 – 2021/22
Q11 (PREDICTION AND CAUSAL INFERENCE)
This question has five parts, (a)-(e). Answer ALL five parts.
A central subject in labour economics is what determines wages. These relationships
include several important areas of research such as returns to education (how do
salaries change for an additional year of schooling), wage premia (gender pay gaps,
socioethnic differences in wages), and union wage effects. Unions play an important
role in enabling collective bargaining and are a main focus of this question.
A panel dataset of 4,360 observations over the period 1980-1987 (N=4360, T=8,
i=545) that can be used for the analysis of wage determination is summarised below.
The dataset includes several variables such as wage, years of schooling, union
membership (union=1 if member, =0 otherwise), and years of labour market
experience.
The variables in this dataset include:
Variable Obs Description
id 4,360 Person identifier
year 4,360 1980 to 1987
wage 4,360 Hourly wage in £
ln_wage 4,360 Log of hourly wage in £
educ 4,360 Years of schooling, range=[3-16], (discrete numerical)
exper 4,360 Labour market experience, range=[0-18], (discrete numerical)
union 4,360 Dummy=1 if respondent is a union member, =0 otherwise
The dataset is declared as a panel in Stata:
. xtset id year
panel variable: id (strongly balanced)
time variable: year, 1980 to 1987
delta: 1 unit
Summary statistics for all the variables of interest are reported below using the xtsum
command:
. xtsum wage ln_wage educ exper union
Variable | Mean Std. Dev. Min Max | Observations
-----------------+--------------------------------------------+----------------
wage overall | 4.320998 2.337624 .020368 41.97815 | N = 4360
between | 1.792748 1.097602 20.70058 | n = 545
within | 1.5019 -11.98915 25.59857 | T = 8
| |
ln_wage overall | 1.334436 .5326094 -3.89379 3.737149 | N = 4360
between | .3907468 .0186327 2.859462 | n = 545
within | .3622636 -2.781912 2.889976 | T = 8
| |
educ overall | 11.76697 1.746181 3 16 | N = 4360
between | 1.747585 3 16 | n = 545
within | 0 11.76697 11.76697 | T = 8
| |
exper overall | 6.514679 2.825873 0 18 | N = 4360
between | 1.654918 3.5 14.5 | n = 545
within | 2.291551 3.014679 10.01468 | T = 8
| |
union overall | .2440367 .4295639 0 1 | N = 4360
between | .3294467 0 1 | n = 545
within | .2759787 -.6309633 1.119037 | T = 8
C29IE
Page 5 of 10
Semester 1 – 2021/22
(a) From this table you can determine which of these variables are time-varying
variables. List the variables that don’t change over time and explain why this is
or isn’t expected.
A junior researcher from an international organisation is building a causal model
of wages using this panel data. The first attempt at the model includes a quadratic
specification for experience, which is common practice in the literature. The
following econometric model is used:
ln () = 0 + 1 + 2 + 32 + 4 + ,
= 1, … ,545, = 1, … ,8
The model is first estimated using OLS. The Stata regression output of the OLS
estimation is reported below. Note that traditional (“unrobust”) standard errors are
reported.
. reg ln_wage educ c.exper##c.exper i.union
Source | SS df MS Number of obs = 4,360
-------------+---------------------------------- F(4, 4355) = 218.64
Model | 206.78755 4 51.6968875 Prob > F = 0.0000
Residual | 1029.74207 4,355 .236450532 R-squared = 0.1672
-------------+---------------------------------- Adj R-squared = 0.1665
Total | 1236.52962 4,359 .283672773 Root MSE = .48626
---------------------------------------------------------------------------------
ln_wage | Coef. Std. Err. t P>|t| [95% Conf. Interval]
----------------+----------------------------------------------------------------
educ | .1028767 .0046288 22.23 0.000 .0938018 .1119515
exper | .0992749 .0100752 9.85 0.000 .0795224 .1190274
|
c.exper#c.exper | -.0031733 .0007127 -4.45 0.000 -.0045705 -.001776
|
1.union | .1734621 .0171735 10.10 0.000 .1397932 .207131
_cons | -.4051742 .0632968 -6.40 0.000 -.5292681 -.2810802
---------------------------------------------------------------------------------
. margins, at(exper=(5 15))
Predictive margins Number of obs = 4,360
Model VCE : OLS
Expression : Linear prediction, predict()
1._at : exper = 5
2._at : exper = 15
------------------------------------------------------------------------------
| Delta-method
| Margin Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
_at |
1 | 1.264747 .0091505 138.22 0.000 1.246807 1.282686
2 | 1.622841 .0488856 33.20 0.000 1.527 1.718681
------------------------------------------------------------------------------
(b) Comment on the specification and overall results. Include in your answer
interpretations of the magnitudes of the estimated coefficients for education and
union membership. Be sure also to include in your discussion an explanation of
the precision of these estimates. Finally, explain why the junior researcher
probably should not use OLS with this dataset.
(c) Using the econometric model presented before, obtain a general expression for
the marginal effect of experience. (Hint: partial differentiation.) Interpret the
marginal effects reported above from the Stata margins command.
C29IE
Page 6 of 10
Semester 1 – 2021/22
Aware of the inadequacy of the OLS estimates, the researcher opts for a fixed effects
(FE) specification in which she reports cluster-robust standard errors.
The fixed effects regression results are presented below:
. xtreg ln_wage educ c.exper##c.exper i.union, fe vce(cluster id)
note: educ omitted because of collinearity
Fixed-effects (within) regression Number of obs = 4,360
Group variable: id Number of groups = 545
R-sq: Obs per group:
within = 0.1767 min = 8
between = 0.0002 avg = 8.0
overall = 0.0562 max = 8
F(3,544) = 143.24
corr(u_i, Xb) = -0.1318 Prob > F = 0.0000
(Std. Err. adjusted for 545 clusters in id)
---------------------------------------------------------------------------------
| Robust
ln_wage | Coef. Std. Err. t P>|t| [95% Conf. Interval]
----------------+----------------------------------------------------------------
educ | 0 (omitted)
exper | .1219331 .0105931 11.51 0.000 .1011248 .1427414
|
c.exper#c.exper | -.004482 .0006872 -6.52 0.000 -.0058319 -.0031322
|
1.union | .0833338 .0229689 3.63 0.000 .0382153 .1284524
_cons | .7457495 .0368206 20.25 0.000 .6734215 .8180775
----------------+----------------------------------------------------------------
sigma_u | .40345324
sigma_e | .35149126
rho | .56850434 (fraction of variance due to u_i)
---------------------------------------------------------------------------------
. margins, at(exper=(5 15))
Predictive margins Number of obs = 4,360
Model VCE : Robust
Expression : Linear prediction, predict()
1._at : exper = 5
2._at : exper = 15
------------------------------------------------------------------------------
| Delta-method
| Margin Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
_at |
1 | 1.263701 .0048389 261.15 0.000 1.254217 1.273185
2 | 1.586629 .0415596 38.18 0.000 1.505174 1.668084
------------------------------------------------------------------------------
(d) Explain in words or with algebra how the fixed effects estimator works. Explain
why the variable years of schooling drops out of the FE estimation. Explain why
it is a good idea to use cluster-robust standard errors here.
(e) Interpret the results and the reported coefficients. Can we interpret the coefficient
on union as causal? Does this mean that joining a union increases salaries? If
so, through which mechanisms? Why does the union premium decrease,
compared to the reported coefficient in the OLS specification?
C29IE
Page 7 of 10
Semester 1 – 2021/22
Q12 (TIME SERIES)
This question has three parts, (a) (seasonality), (b) (stationarity), and (c) (forecasting).
Answer ALL three parts. The question is about “Okun’s Law”. Okun’s Law is the
empirical relationship between unemployment and GDP growth. It suggests that an
increase in unemployment rate is associated with a fall in GDP growth. Using the US
economy, we investigate some characteristics of this relationship.
a. (Seasonality)
When studying Okun’s Law, econometricians tend to use seasonally adjusted
data, so we start by looking at seasonality. The figures below show seasonally
unadjusted quarterly GDP and quarterly e-commerce sales in the US. Please
describe seasonality in your own words and explain your observations from the
two figures. What can you conclude about the seasonality of overall economic
activity and of e-commerce in particular by looking at the figures below?
C29IE
Page 8 of 10
Semester 1 – 2021/22
b. (Stationarity)
log of GDP in period t
= − −1
= − −1 (first difference of unemployment in period t)
: time index in period , increases from 1, … ,
Before we proceed, we plot our GDP series to inspect it.
C29IE
Page 9 of 10
Semester 1 – 2021/22
i. Based on the graph above of log GDP and its first-difference, do you see
any visual evidence of time trends? Explain.
ii. Explain what potential problems can arise when predicting using the level
of log GDP and why you might want to use the first difference of log GDP
instead. Explain how you can use a Dickey-Fuller test to answer the
question of whether to use the level or first difference. What are the null
and alternative hypotheses of the test?
iii. Below are the results from applying a Dickey-Fuller test to this dataset.
Comment on the results.
. dfuller lgdp, lags(4) regress
Augmented Dickey-Fuller test for unit root Number of obs = 81
---------- Interpolated Dickey-Fuller ---------
Test 1% Critical 5% Critical 10% Critical
Statistic Value Value Value
------------------------------------------------------------------------------
Z(t) -1.019 -3.537 -2.905 -2.588
MacKinnon approximate p-value for Z(t) = 0.7463
------------------------------------------------------------------------------
D.lgdp | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
lgdp |
L1. | -.0089703 .008803 -1.02 0.311 -.0265066 .0085661
LD. | -.1743878 .1155175 -1.51 0.135 -.4045104 .0557348
L2D. | -.0371176 .1188938 -0.31 0.756 -.2739662 .199731
L3D. | -.0401303 .1208881 -0.33 0.741 -.2809517 .200691
L4D. | -.2259624 .1411382 -1.60 0.114 -.507124 .0551992
|
_cons | .1003119 .0856033 1.17 0.245 -.0702187 .2708425
------------------------------------------------------------------------------
C29IE
Page 10 of 10
Semester 1 – 2021/22
c. (Forecasting)
To forecast US GDP for 2021:3 (YYYY:Q, so the 3rd quarter of 2021), you
estimate an AR(4) and an ADL(4,1) model for the sample period 2000:1 to
2021:2. The results are as follows. Standard errors are in parentheses.
Model 1:
AR(4)
Model 2:
ADL(4,1)
L.dlgdp -0.168 1.198
(0.115) (0.278)
L2.dlgdp -0.025 0.147
(0.118) (0.107)
L3.dlgdp -0.024 -0.176
(0.120) (0.107)
L4.dlgdp -0.191 -0.206
(0.137) (0.118)
L.dunemp
0.019
(0.004)
_cons 0.013 0.000
(0.003) (0.004)
N 81 81
rmse 0.016 0.014
In addition, you have the following information:
Quarter lgdp dlgdp unemp dunemp
2020:3 9.959 0.082 8.8 -4.3
2020:4 9.975 0.016 6.8 -2
2021:1 10.001 0.026 6.2 -0.6
2021:2 10.032 0.031 5.9 -0.3
2021:3 . . 5.1 -0.8
i. Predict GDP growth in 2021:3 using both models given this information.
Comparing the forecasts you obtained, comment on whether your findings are in
line with Okun’s Law. (Hint: not all “laws” in economics are obeyed!)
ii. Construct the 95% forecast interval around your predictions. Hint: you can use the
RMSE as an approximation for the mean squared forecast error (MSFE).
iii. Which of the two models has a better predictive performance? Explain. Can you
think of any reason(s) for the two predictions to be noticeably different from each
other?
Page 1 of 10
Semester 1 – 2021/22
SCHOOL OF SOCIAL SCIENCES
Edinburgh Business School
Course Name: INTRODUCTORY ECONOMETRICS
Course Code: C21IE
SEMESTER ONE – 2021/22
INSTRUCTIONS
Section A – Answer BOTH questions
Section B – Answer any FIVE questions
Section C – Answer only ONE question
C29IE
Page 2 of 10
Semester 1 – 2021/22
Section A (THEORY) (26 marks)
Q1. This question has two parts, (a) and (b). Answer ALL parts.
(a) What is a confidence interval and how should it be interpreted? Explain how we
can use our knowledge of the sampling distribution of the OLS estimator ̂ to
construct a confidence interval.
(b) What is “omitted variable bias”? What is a “control variable”? Explain how
control variables can be used to address the failure of Least Squares Assumption
#1 for Causal Inference, i.e., the failure of the assumption that E(ui|Xi)=0.
Section B: ANSWER ANY 4 OF THE FOLLOWING 9 QUESTIONS (24 marks, 6
marks per answer)
Q2: Least Squares Assumption #1 for Prediction is: “The out-of-sample observation
(XOOS,YOOS) is drawn from the same distribution as the estimation sample
(Xi,Yi), i = 1,…, n.” What are the consequences for prediction based on OLS if
this assumption fails?
Q3: What are “robust standard errors” and why do we use them?
Q4: Explain what is meant by pseudo-out-of-sample forecast.
Q5: What is the Lasso and when should we use it?
Q6: Explain what it means for a time series to be stationary.
Q7: Define AIC (Akaike Information Criterion) and discuss how it can be used to
select the number of lags in a time series regression.
Q8: What is a DAG and how can it be used for causal inference? Provide an
example of a simple DAG.
Q9: What is the method of “differences-in-differences”? Give an example of its use.
Q10: Given an example of a measure of “goodness of fit”, and explain what it means
and how it can be used.
C29IE
Page 3 of 10
Semester 1 – 2021/22
Section C: ANSWER ANY 1 OF THE FOLLOWING 2 QUESTIONS (50 marks)
Q11: Prediction and causal inference
Q12: Time series
(Section C questions follow below.)
C29IE
Page 4 of 10
Semester 1 – 2021/22
Q11 (PREDICTION AND CAUSAL INFERENCE)
This question has five parts, (a)-(e). Answer ALL five parts.
A central subject in labour economics is what determines wages. These relationships
include several important areas of research such as returns to education (how do
salaries change for an additional year of schooling), wage premia (gender pay gaps,
socioethnic differences in wages), and union wage effects. Unions play an important
role in enabling collective bargaining and are a main focus of this question.
A panel dataset of 4,360 observations over the period 1980-1987 (N=4360, T=8,
i=545) that can be used for the analysis of wage determination is summarised below.
The dataset includes several variables such as wage, years of schooling, union
membership (union=1 if member, =0 otherwise), and years of labour market
experience.
The variables in this dataset include:
Variable Obs Description
id 4,360 Person identifier
year 4,360 1980 to 1987
wage 4,360 Hourly wage in £
ln_wage 4,360 Log of hourly wage in £
educ 4,360 Years of schooling, range=[3-16], (discrete numerical)
exper 4,360 Labour market experience, range=[0-18], (discrete numerical)
union 4,360 Dummy=1 if respondent is a union member, =0 otherwise
The dataset is declared as a panel in Stata:
. xtset id year
panel variable: id (strongly balanced)
time variable: year, 1980 to 1987
delta: 1 unit
Summary statistics for all the variables of interest are reported below using the xtsum
command:
. xtsum wage ln_wage educ exper union
Variable | Mean Std. Dev. Min Max | Observations
-----------------+--------------------------------------------+----------------
wage overall | 4.320998 2.337624 .020368 41.97815 | N = 4360
between | 1.792748 1.097602 20.70058 | n = 545
within | 1.5019 -11.98915 25.59857 | T = 8
| |
ln_wage overall | 1.334436 .5326094 -3.89379 3.737149 | N = 4360
between | .3907468 .0186327 2.859462 | n = 545
within | .3622636 -2.781912 2.889976 | T = 8
| |
educ overall | 11.76697 1.746181 3 16 | N = 4360
between | 1.747585 3 16 | n = 545
within | 0 11.76697 11.76697 | T = 8
| |
exper overall | 6.514679 2.825873 0 18 | N = 4360
between | 1.654918 3.5 14.5 | n = 545
within | 2.291551 3.014679 10.01468 | T = 8
| |
union overall | .2440367 .4295639 0 1 | N = 4360
between | .3294467 0 1 | n = 545
within | .2759787 -.6309633 1.119037 | T = 8
C29IE
Page 5 of 10
Semester 1 – 2021/22
(a) From this table you can determine which of these variables are time-varying
variables. List the variables that don’t change over time and explain why this is
or isn’t expected.
A junior researcher from an international organisation is building a causal model
of wages using this panel data. The first attempt at the model includes a quadratic
specification for experience, which is common practice in the literature. The
following econometric model is used:
ln () = 0 + 1 + 2 + 32 + 4 + ,
= 1, … ,545, = 1, … ,8
The model is first estimated using OLS. The Stata regression output of the OLS
estimation is reported below. Note that traditional (“unrobust”) standard errors are
reported.
. reg ln_wage educ c.exper##c.exper i.union
Source | SS df MS Number of obs = 4,360
-------------+---------------------------------- F(4, 4355) = 218.64
Model | 206.78755 4 51.6968875 Prob > F = 0.0000
Residual | 1029.74207 4,355 .236450532 R-squared = 0.1672
-------------+---------------------------------- Adj R-squared = 0.1665
Total | 1236.52962 4,359 .283672773 Root MSE = .48626
---------------------------------------------------------------------------------
ln_wage | Coef. Std. Err. t P>|t| [95% Conf. Interval]
----------------+----------------------------------------------------------------
educ | .1028767 .0046288 22.23 0.000 .0938018 .1119515
exper | .0992749 .0100752 9.85 0.000 .0795224 .1190274
|
c.exper#c.exper | -.0031733 .0007127 -4.45 0.000 -.0045705 -.001776
|
1.union | .1734621 .0171735 10.10 0.000 .1397932 .207131
_cons | -.4051742 .0632968 -6.40 0.000 -.5292681 -.2810802
---------------------------------------------------------------------------------
. margins, at(exper=(5 15))
Predictive margins Number of obs = 4,360
Model VCE : OLS
Expression : Linear prediction, predict()
1._at : exper = 5
2._at : exper = 15
------------------------------------------------------------------------------
| Delta-method
| Margin Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
_at |
1 | 1.264747 .0091505 138.22 0.000 1.246807 1.282686
2 | 1.622841 .0488856 33.20 0.000 1.527 1.718681
------------------------------------------------------------------------------
(b) Comment on the specification and overall results. Include in your answer
interpretations of the magnitudes of the estimated coefficients for education and
union membership. Be sure also to include in your discussion an explanation of
the precision of these estimates. Finally, explain why the junior researcher
probably should not use OLS with this dataset.
(c) Using the econometric model presented before, obtain a general expression for
the marginal effect of experience. (Hint: partial differentiation.) Interpret the
marginal effects reported above from the Stata margins command.
C29IE
Page 6 of 10
Semester 1 – 2021/22
Aware of the inadequacy of the OLS estimates, the researcher opts for a fixed effects
(FE) specification in which she reports cluster-robust standard errors.
The fixed effects regression results are presented below:
. xtreg ln_wage educ c.exper##c.exper i.union, fe vce(cluster id)
note: educ omitted because of collinearity
Fixed-effects (within) regression Number of obs = 4,360
Group variable: id Number of groups = 545
R-sq: Obs per group:
within = 0.1767 min = 8
between = 0.0002 avg = 8.0
overall = 0.0562 max = 8
F(3,544) = 143.24
corr(u_i, Xb) = -0.1318 Prob > F = 0.0000
(Std. Err. adjusted for 545 clusters in id)
---------------------------------------------------------------------------------
| Robust
ln_wage | Coef. Std. Err. t P>|t| [95% Conf. Interval]
----------------+----------------------------------------------------------------
educ | 0 (omitted)
exper | .1219331 .0105931 11.51 0.000 .1011248 .1427414
|
c.exper#c.exper | -.004482 .0006872 -6.52 0.000 -.0058319 -.0031322
|
1.union | .0833338 .0229689 3.63 0.000 .0382153 .1284524
_cons | .7457495 .0368206 20.25 0.000 .6734215 .8180775
----------------+----------------------------------------------------------------
sigma_u | .40345324
sigma_e | .35149126
rho | .56850434 (fraction of variance due to u_i)
---------------------------------------------------------------------------------
. margins, at(exper=(5 15))
Predictive margins Number of obs = 4,360
Model VCE : Robust
Expression : Linear prediction, predict()
1._at : exper = 5
2._at : exper = 15
------------------------------------------------------------------------------
| Delta-method
| Margin Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
_at |
1 | 1.263701 .0048389 261.15 0.000 1.254217 1.273185
2 | 1.586629 .0415596 38.18 0.000 1.505174 1.668084
------------------------------------------------------------------------------
(d) Explain in words or with algebra how the fixed effects estimator works. Explain
why the variable years of schooling drops out of the FE estimation. Explain why
it is a good idea to use cluster-robust standard errors here.
(e) Interpret the results and the reported coefficients. Can we interpret the coefficient
on union as causal? Does this mean that joining a union increases salaries? If
so, through which mechanisms? Why does the union premium decrease,
compared to the reported coefficient in the OLS specification?
C29IE
Page 7 of 10
Semester 1 – 2021/22
Q12 (TIME SERIES)
This question has three parts, (a) (seasonality), (b) (stationarity), and (c) (forecasting).
Answer ALL three parts. The question is about “Okun’s Law”. Okun’s Law is the
empirical relationship between unemployment and GDP growth. It suggests that an
increase in unemployment rate is associated with a fall in GDP growth. Using the US
economy, we investigate some characteristics of this relationship.
a. (Seasonality)
When studying Okun’s Law, econometricians tend to use seasonally adjusted
data, so we start by looking at seasonality. The figures below show seasonally
unadjusted quarterly GDP and quarterly e-commerce sales in the US. Please
describe seasonality in your own words and explain your observations from the
two figures. What can you conclude about the seasonality of overall economic
activity and of e-commerce in particular by looking at the figures below?
C29IE
Page 8 of 10
Semester 1 – 2021/22
b. (Stationarity)
log of GDP in period t
= − −1
= − −1 (first difference of unemployment in period t)
: time index in period , increases from 1, … ,
Before we proceed, we plot our GDP series to inspect it.
C29IE
Page 9 of 10
Semester 1 – 2021/22
i. Based on the graph above of log GDP and its first-difference, do you see
any visual evidence of time trends? Explain.
ii. Explain what potential problems can arise when predicting using the level
of log GDP and why you might want to use the first difference of log GDP
instead. Explain how you can use a Dickey-Fuller test to answer the
question of whether to use the level or first difference. What are the null
and alternative hypotheses of the test?
iii. Below are the results from applying a Dickey-Fuller test to this dataset.
Comment on the results.
. dfuller lgdp, lags(4) regress
Augmented Dickey-Fuller test for unit root Number of obs = 81
---------- Interpolated Dickey-Fuller ---------
Test 1% Critical 5% Critical 10% Critical
Statistic Value Value Value
------------------------------------------------------------------------------
Z(t) -1.019 -3.537 -2.905 -2.588
MacKinnon approximate p-value for Z(t) = 0.7463
------------------------------------------------------------------------------
D.lgdp | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
lgdp |
L1. | -.0089703 .008803 -1.02 0.311 -.0265066 .0085661
LD. | -.1743878 .1155175 -1.51 0.135 -.4045104 .0557348
L2D. | -.0371176 .1188938 -0.31 0.756 -.2739662 .199731
L3D. | -.0401303 .1208881 -0.33 0.741 -.2809517 .200691
L4D. | -.2259624 .1411382 -1.60 0.114 -.507124 .0551992
|
_cons | .1003119 .0856033 1.17 0.245 -.0702187 .2708425
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C29IE
Page 10 of 10
Semester 1 – 2021/22
c. (Forecasting)
To forecast US GDP for 2021:3 (YYYY:Q, so the 3rd quarter of 2021), you
estimate an AR(4) and an ADL(4,1) model for the sample period 2000:1 to
2021:2. The results are as follows. Standard errors are in parentheses.
Model 1:
AR(4)
Model 2:
ADL(4,1)
L.dlgdp -0.168 1.198
(0.115) (0.278)
L2.dlgdp -0.025 0.147
(0.118) (0.107)
L3.dlgdp -0.024 -0.176
(0.120) (0.107)
L4.dlgdp -0.191 -0.206
(0.137) (0.118)
L.dunemp
0.019
(0.004)
_cons 0.013 0.000
(0.003) (0.004)
N 81 81
rmse 0.016 0.014
In addition, you have the following information:
Quarter lgdp dlgdp unemp dunemp
2020:3 9.959 0.082 8.8 -4.3
2020:4 9.975 0.016 6.8 -2
2021:1 10.001 0.026 6.2 -0.6
2021:2 10.032 0.031 5.9 -0.3
2021:3 . . 5.1 -0.8
i. Predict GDP growth in 2021:3 using both models given this information.
Comparing the forecasts you obtained, comment on whether your findings are in
line with Okun’s Law. (Hint: not all “laws” in economics are obeyed!)
ii. Construct the 95% forecast interval around your predictions. Hint: you can use the
RMSE as an approximation for the mean squared forecast error (MSFE).
iii. Which of the two models has a better predictive performance? Explain. Can you
think of any reason(s) for the two predictions to be noticeably different from each
other?
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