MSIN0105 Financial Econometrics
2020/2021
Toru Kitagawa and Rui Silva
Mock Exam solutions (Luca Coraggio)
Colors
• Questions are given in blue;
• Answers are given in black;
• Additional remarks (not needed for full marks) are given in red
Question 1
Judge if each statement given below is correct or incorrect, providing your reasoning. In these state-
ments, (yi, xi), i = 1, ..., n, are iid observations of a dependent variable yi and a vector of regressors
xi = (1, x1i, . . . , xKi)
1. Let z be a k × 1 random vector, k ≥ 1, following the multivariate normal distribution with mean 0 and
variance-covariance matrix Σ. The squared length of z, z′z, follows the chi-square distribution with the
degree of freedom equal to k.
Answer: The vector z is as follows:
z =

z1
z2
...
zK
 ,
which implies that:
z′z =
[
z1 z2 · · · zK
]

z1
z2
...
zK
 =
K∑
i=1
z2i .
From the above, we see that the squared length z′z is the sum of K normal random variables, squared.
However, a Chi-squared arises from the sum of independent, squared standard normals. Thus, if Σ 6= IK
(the K-dimensional identity matrix), the statement is false.
(Optionally)
The correct statement would be z′Σ−1z ∼ χ2(K). In fact, since Σ is a symmetric, positive definite matrix, it
can be decomposed as Σ = V ΛV ′, where V are the eigenvectors of Σ and are orthogonal (i.e. V V ′ = I) and Λ
is a diagonal matrix containing the eigenvalues of Σ. Call C = V Λ−1/2, then
CC ′ = V Λ− 12Λ− 12V ′ = V Λ−1V ′ = (V ΛV ′)−1 = Σ−1.
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Note that C ′z ∼ N(0, CΣC ′) = N(0, I), since
C ′ΣC = C ′V ΛV ′C = Λ− 12 V ′V︸︷︷︸
I
ΛV ′V︸︷︷︸
I
Λ− 12 = Λ− 12ΛΛ− 12 = I;
thus (C ′z)′ (C ′z) = z′CC ′z = z′Σ−1z ∼ χ2(K).

2. Specifying the linear model as yi = x′iβ + i with E (i|xi) = 0 implies that the regression equation of yi
onto xi is x′iβ.
Answer: The statement is correct, since:
E(yi|xi) = E (x′iβ + i|xi) = E (x′iβ|xi) + E (i|xi) = x′iβ

3. The linear probability model yi = x′iβ + i with yi ∈ {0, 1} has heteroskedastic errors, i.e., V ar(i|xi)
depends on xi.
Answer: The statement is correct. In fact, since P (yi = 1|xi) = x′iβ, we have:
V ar(i|xi) = V ar(yi − x′iβ|xi) = V ar(yi|xi) = E(y2i |xi)− E(yi|xi)2 =
12P (yi = 1|xi) + 02P (yi = 0|xi)− (1P (yi = 1|xi) + 0P (yi = 0|xi))2 =
P (yi = 1|xi)− P (yi = 1|xi)2 = P (yi = 1|xi) (1− P (yi = 1|xi)) =
x′iβ (1− x′iβ)

4. Consider a probit regression model, Pr(yi = 1|xi) = Φ (x′iβ), where Φ (·) is the cumulative distribution
function of the standard normal distribution. The marginal effect with respect to k-th regressor,
∂
∂xki
Pr (yi = 1|xi) for k ∈ {1, . . . ,K}, does not depend on xi.
Answer: The statement is false, because the conditional probability is a non-linear function of x. We can
show this formally:
∂
∂xki
P (yi = 1|xi) = ∂
∂xki
Φ (x′iβ) = φ (x′iβ)βk.

Question 2
Let rt, t = 0, 1, 2, . . . , T , be a sequence of daily log returns of a market price index (e.g., S&P500). To assess
dependence of the daily log returns, consider estimating the following regression equation:
rt = β0 + β1rt−1 + t, (1)
for t = 1, 2, . . . , T . The following table reports the OLS estimates βˆ =
(
βˆ0, βˆ1
)
, and robust standard errors
(robust s.e.), i.e., an estimate for
√
V ar
(
βˆk
)
, k = 0, 1.
Variable OLS estimates robust s.e.
Intercept 0.032 0.015
rt−1 -0.027 0.021
(a) Explain the market efficiency hypothesis for the stock market, and discuss why the market efficiency
hypothesis implies β1 = 0 and {t : t = 1, 2, . . . } being statistically independent over t = 1, . . . , T .
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Answer: The market efficiency hypothesis says the future asset price cannot be predicted based on the
available information up to now due to the quick adjustment of the asset price in response to any news
available to the investors. Under this hypothesis, we have that (log) prices follow a random walk:
pt = pt−1 + t.
But this implies that the log returns can be written as:
rt = pt − pt−1 = t, t⊥s, t 6= s.
Thus, when regressing rt on past values,
rt = β0 + β1rt−1 + t,
we should find that β1 = 0.

(b) Based on the table of the estimation result reported above, perform a hypothesis testing for the null
hypothesis that β1 = 0. You may use the fact that the 97.5% quantile of the standard normal distribution
is 1.96.
Answer: We know that (under appropriate assumptions):
√
n
(
βˆOLS − β
)
→d N(0,Ω)
From this, it follows that:
√
n
(
βˆOLS1 − β1
)
→d N(0, w22) =⇒
√
n
(
βˆOLS1 − β1
)
√
w22
→d N(0, 1),
(where w22 is the element of position 2, 2 in matrix Ω). Note that w22 needs to be estimated, so we replace it
with ωˆ22; we obtain the T-statistic:
√
n
(
βˆOLS1 − β1
)
√
wˆ22
= βˆ
OLS
1 − β1
s.e.
(
βˆOLS1
)
︸ ︷︷ ︸√
wˆ22
n
= T →d N(0, 1).
We can compute the realized value of the T-statistic, t, by replacing the values from the table above:
βˆOLS1 = −0.027; s.e.
(
βˆOLS1
)
= 0.021. Thus, under the null (H0 : β1 = 0), we have:
lim
n→∞P
(
|T | ≥
∣∣∣−0.027− 00.021 ∣∣∣
)
= lim
n→∞P (|T | ≥ 1.29) > limn→∞P (|T | ≥ 1.96)
The obtained probability value is higher than the pre-specified probability value to reject the null. Thus, we
do not reject the null hypothesis (at least at 5% confidence level).

(c) Does the market efficiency hypothesis allow β0 to be different from zero? Explain your answer.
Answer: Yes, it does. In this case, it is still not possible to predict future prices with past prices, but the
prices follows a random walk with drifts:
pt = pt−1 + β0 + t =⇒ rt = pt − pt−1 = β0 + t.
This can happen when the available safe asset has an interest rate greater than 0 (which implies β0 > 0).
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(d) The table of the estimation result above reports robust standard errors rather than homoskedastic
standard errors. Justify why we prefer the robust standard errors in the current context.
Answer: We should use the robust standard errors if the variance of t depends on the regressor rt−1. The
stock returns often show heteroskedastic errors, i.e., the variance of the returns depend on |rt−1|. Hence, we
prefer the robust s.e. to perform inference for the coefficients.

(e) We want to perform a hypothesis test for the joint null hypothesis β0 = β1 = 0. Can you perform the
test based on the estimation result reported in the table above? Explain your answer.
Answer: The test cannot be performed with just the information in the table above. We would need also to
take into account the asymptotic covariance of β0 and β1 in such a test, but this information is missing in the
table above. An appropriate test could be constructed via the Wald statistic as follows
W = n
(
RβˆOLS − r
)′ [
RΩˆR′
] (
RβˆOLS − r
)
∼ χ2(2),
where
R =
[
1 0
0 1
]
; r =
[
0
0
]
.
The matrix Ωˆ is not obtained from the table above (we could obtain only its main diagonal).

Question 3
Consider estimating a demand equation for gasoline
yi = βppi + w′iβw + ui
where yi is the aggregated quantity of gasoline consumed in market i, pi is unit price in market i, wi is a
vector of observable characteristics in market i, and ui is unobserved demand shock.
Assume that each market is geographically separated and the sample consists of iid observations of (yi, pi, wi),
i = 1, . . . , n. Throughout this question, we specify observations of price pi to be endogenous and observable
characteristics wi to be exogenous.
(a) What are reasons that we consider pi to be an endogenous variable? Discuss.
Answer: The issue with this specification is *simultaneity*. Assuming that the price of gasoline is determined
as the equilibrium price, i.e. the intersection point of demand and supply for gasoline, we have:
Demand: yi = βppi + ω′iβw + ui
Supply: yi = γppi + ω′iγw + vi
Solving for the price, we obtain (assume that βp 6= γp):
βppi + ω′iβw + ui = γppi + ω′iγw + vi ⇐⇒ pi =
1
βp − γp [ω
′
i (γw − βw) + vi − ui] ,
which clearly shows that Cov(pi, ui) 6= 0 (i.e. the price is correlated with demand shocks). Thus, the price is
endogenous in the demand equation.

(b) In order to estimate the demand equation by the two-stage least squares (2SLS), propose a candidate
of instrumental variables, and discuss why you think the proposed instrument is valid in the current
context.
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Answer: A valid instrument, z1i, would be one such that it affects demanded quantity of gasoline only
through its effects on prices (rank condition); but it should also be uncorrelated with demand shocks (exclusion
restriction). Such a instrument could be, for example, a variable that measures the cost of importing oil
such as geographical or time distance between the market and nearby OPEC countries or oil field. The rank
condition will hold because z1i can shift up or down the supply function through change in the price of inputs,
so z1i and the gasoline price should be dependent. On the other hand, it would be reasonable to think z1i is
not dependent on unobservable demand shock in market i, so the exclusion restriction seems to be valid.

(c) In implementing 2SLS, how do you check whether the instrument is weak or not? State a procedure for
it.
Answer: For the instrument not to be weak, it means that the impact on the instrumented variable should
not be weak. That is, it should be possible to predict the endogenous variable via the instruments (i.e. the
correlation between the two should not be too weak). Consider the first stage regression:
pi = pizz1i + ω′ipiω + ηi,
when the instrument is weak, then coefficient piz ≈ 0. This can be tested by estimating the first stage
regression and then performing an F-test on the hypothesis that instruments’ coefficients are equal to 0. The
heuristics is that if F-stat > 10, then the instrument(s) is(are) valid and is(are) said not to be weak.

(d) State what problems the weak instrument causes.
Answer: Weak instruments causes the 2SLS estimator, βˆ2SLS, to be heavily biased (potentially more than
OLS). Also, inference becomes unreliable because, even in large samples, the sampling distribution of the
2SLS estimator may be far from normal.
(optionally) One possibility to cope with weak instruments is to use the Limited Information Maximum
Likelihood method, which is median unbiased and its sampling distribution is centered on the true value.
However, the sampling distribution has fat tails, so even if on average it is better than 2SLS, in the sample at
hand the LIML estimator may be far from the truth.

Question 4
After finishing your program at UCL you are offered a job at you favourite investment bank. Your first client
is a firm that is planning to diversify their business, that is, the firm is planning to expand its operations
from a single industrial segment to multiple industries. Since this is an important strategic shift for the firm,
the CEO of the company has hired your bank to advise on this decision.
Part 1: Diversification discount
The first thing you do is show your client evidence on the relationship between corporate diversification and
firm value.
You present to your client a table from Berger and Ofek’s paper in the Journal of Financial Economics titled
“Diversification’s effect on firm value.” The dependent variable in this table is the ratio of the value of a
diversified firm to the value of a portfolio of stand alone firms operating in the same industry as the diversified
firm. This variable intends to capture whether being diversified is associated with superior or inferior value
than similar pure play firms.
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(a) What is the name of this type of analysis?
Answer:
The table represented above is just what you also showed in your analysis in the CourseWork. The name of
this analysis is simply *multivariate OLS* or simply *OLS*. (Both answers are fine)
(b) The main explanatory variable you want to discuss with your client is Multi-segment indicator. This is
an indicator variable that takes the value of one if the firm is diversified and takes the value of zero
otherwise. Given the coefficients in this table, what can you tell your client about the relationship
between diversification and firm value? Note that the superscript d denotes statistical significance at
the one percent level.
Answer: On average diversified firms trade at a discount relative to their stand-alone counterparts. In fact,
the value of the coefficient is negative, indicating that, ceteris paribus, diversified firms show a lower excess
value with respect to not-diversified ones. With p-values equal to zero, the results are statistically significant
at the 1% level. Moreover, in terms of economic significance, the estimates imply "conglomerate discounts"
on the order of 12.7 to 15.2 percent of stand-alone valuations (according to the definition of the dependent
variable; note that the dependent value is defined in logarithms).
(c) Your client asks you whether you are confident that the true coefficients associated with the variable
Multi-segment indicator are not zero. Please provide an explanation that justifies your answer.
Answer: Yes. What your client is asking is, effectively, a significance test where the null hypothesis to be
tested is H0 : βMSI = 0 (MSI reads as "Multi-segment indicator"). This can be easily conducted with the
information in the table above. Note that the values in parentheses are the p-values associated with the
coefficients. With a p-value of (roughly) 0, we know that, were H0 true, the probability of having estimated
the MSI coefficients above would have been (roughly) 0. In fact, remember that p-value tells the probability of
obtaining an estimate greater (in absolute value) than the observed one, under H0: PH0(|T | ≥ |t|). Thus, were
the true coefficients = 0, we would have never estimated the coefficients we estimated (i.e. −0.127, −0.144
and −0.152). Thus, the true coefficients "must" be different from 0. We are really comfortable – given the
table above – saying to our client that the true relationship is strictly negative, even if it may be slightly
different from our point-estimates (why strictly negative? Because, had the confidence interval contained the
value 0, we would not have rejected the null).
Part 2: Productivity and diversification
Given these intriguing facts, your client asks you about any information you may have regarding the association
between diversification and productivity. After all, a firm’s market value may not capture the value it produces
for its stakeholders (e.g., customers, employees, suppliers). You turn to a study by Scholar in the Journal of
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Finance in 2002 titled “Effects of Corporate Diversification on Productivity.” You focus your attention on
table 2, reproduced below.
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(a) In this case, the dependent variable is total factor productivity (a measure of the ability of a production
plant to transform inputs into output) and the main explanatory variable that you focus on is “Two-digit
diversification,” an indicator variable that takes the value of one if the firm is diversified into several
(two-digit) industries, and takes the value of zero for undiversified firms.
Focusing on columns 1 and 3, what do you conclude? Is corporate diversification good or bad for
productivity?
Answer: Focusing on columns 1 and 3 of table II, we conclude that diversification as a positive impact on
production plant’s profitability (on average). That is plants that are owned by diversified firms are more
productive than plants owned by focused firms. These estimates are statistically significant at the 1% level in
column 1 and at the 5% level in column 3, indicating that zero is outside of the 95% confidence interval for
both regression models.
(b) Why are the estimated coefficients different between column 1 and column 3? Which one do you think
is more reliable? Why?
Answer: The estimates differ because the specification of the regression model. Column 1 has a more
parsimonious specification, where the only other regressor is plant’s "Age". Column 3, on the other hand,
adds controls for segment and firm’s sizes. Given that productivity may be determined by things other than
the degree of corporate diversification, failing to include relevant variables in the regression model could
lead to omitted variables bias in the coefficient of "Two-digit diversification". Therefore, the specification of
column 3, adding size controls is probably more reliable in capturing the impact of corporate diversification
on productivity.
(Optionally)
However, note that by constructing approximate confidence intervals for the two coefficients (i.e. βˆ ± 2SE(βˆ)),
we have:
Column 1: [0.034, 0.094] ,
Column 3: [0.002, 0.066] ;
so that the difference in the estimates may be just an artifact from the current sample (i.e. the two specifications
might not be telling different stories).
(c) Why is the estimated coefficient in column 5 fundamentally different from those in columns 1 and 3?
Please explain.
Answer: The main (important) difference with column 5 is that it includes fixed effects controls at the
plant level. That this, the coefficient of "Two-digit diversification" in column 5 is capturing the within-plant
variability (i.e. the variability across years) of the TFP. We are comparing the total factor productivity of
the same plant across years. In the case of the variable "Two-digit diversification", this specification shows
that within the same plant, years when the plant was owned by a diversified firm are associated with lower
productivity than when the plant is owned by a stand alone firm. On the other hand, for columns 1 and 3,
the captured variation in TFP is both across years and plants, showing that diversification is associated with
an increment of TFP (across years and production plants).
Part 3: The value of diversification in bad times
The final question your client asks you is whether the value of being diversified varies over time? In particular,
he asks: do large diversified firms perform better, worse, or the same during crisis as they do in normal times?
You search the literature and find the following table on the value of diversification during crisis from
Kuppuswamy and Villalonga in the 2016 issue of Management Science.
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10
(a) In this table the dependent variable is again firm value. To explain firm value, in addition to a
diversification dummy variable, this table also reports coefficients associated with dummy variables for
crisis periods divided by the start of the crisis (dummy variable “Early crisis”) and the later part of the
crisis (dummy variable “Late crisis”).
Reading this table, what can you say about the correlation between being diversified and firm value
outside of a crisis period?
Answer:
When the crisis indicators are zero, the impact of diversification is given by the coefficient on the variable
"Diversified" alone. Being this coefficient negative and statistically significant (three stars next to the coefficient
typically indicates a 1% significance level), we conclude that the impact of diversification outside crises is
negative on firm’s value. That is, outside crises, diversified firms are associated with a lower value than
non-diversified ones.
(Optionally)
If we assume the firm’s value to be defined similarly to what is done in part 1, we can further say that being
diversified is associated with a 14% lower firm value.
(b) What is the impact of being diversified during a crisis? Given the information in this table, how do you
answer the question from your client?
Answer: To quantify the impact of being diversified on firm value during crisis, we need to sum the coefficient
associated with the variable "Diversified" with that of the interactions between "Diversified" and the crisis
dummies. In fact, our regression model is like this:
yi = βdDi + βcCi + βdcDiCi + x′iβ + i,
where for simplicity, we use a unique indicator variable for the crisis period C, and D is the dummy for
diversification. Then, the marginal effect of being diversified is:
∂
∂D
E(yi|D,C, x) = ∂
∂D
E(yi|D,C, x)βdDi + βcCi + βdcDiCi + x′iβ = βd + βdcCi
(this also motivates the previous answer). Then, in this case, assuming that the "early crisis" dummy and the
"late crisis" dummy are mutually exclusive (they do not take value 1 contemporaneously; which is reasonable),
we have:
Effect of being diversified in Early Crises: (−0.14 + 0.07) = −0.07
Effect of being diversified in Late Crises: (−0.14 + 0.09) = −0.05
Thus, during crises, the diversified firms do a lot better (in terms of firm value) than outside crises. However,
diversification is still worse than non-diversification overall, no matter if in crisis periods or not. In other
words, diversification is associated with a lower "diversification discount" in crisis periods than in normal
times, but there is no evidence of a "diversification premium" even during crisis.
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