EXAM PAPER ECONM1022 ECONOMETRICS
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EXAM PAPER ECONM1022
ECONOMETRICS
2022
Part A [25 marks]
Let y be an outcome variable and z = (1, z1 , z2 , z3 ) a vector of regressors. We have a large iid sample with n observations of y and z. An OLS regression of y on z using this data sample gives the estimate = ╱0 , 1 , 2 , 3、\
1. Show that is equal to (X\X)L1X\ y , where X and y are data matrices that you need to specify. What condition is needed for the OLS estimator to be well defined? [4 marks]
From now on, we consider the linear model y = g0 + g1z1 + g2z2 + g3z3 + t where g = (g0 , g1 , g2 , g3 )\ is a vector of causal parameters and t is an unobserved error term. We assume that the exogeneity condition holds: E(t|z) = 0.
2. Show that is an unbiased estimate of g . [4 marks]
3. How would you test that z1 , z2 and z3 all have the same effect on y? [4 marks]
4. (a) Show that E(tz) = 0. [2 marks]
(b) How can you approximate E(tz) using the data sample? [2 marks]
(c) Show that the OLS estimator can be characterised as the solution of a moment equation. [2 marks]
5. We now regress y on a constant, z1 and z2 by OLS using the data sample and denote the resulting estimate as = ╱0 , 1 , 2、\ .
(a) How does the R2 from this regression compare with the R2 from the OLS regres-
sion that produced ? [2 marks]
(b) Show that 1 and 2 are consistent estimates of g1 and g2 if E (z3 |1, z1 , z2 ) = 0 or
if g3 = 0. Comment. [5 marks]
Part B [25 marks]
We want to study how income subsidies may improves children’s education outcomes. Let y be the test score (between 0 and 100%) obtained by a child at the end of primary school (for instance a key stage 2 maths exam in the UK). We denote as w the monthly income of a child’s household. The dummy variable d equals 1 if the household received an income subsidy and is equal to 0 otherwise.
We consider the linear model y = g0 + g1 d + g2w + t, where t is an unobserved error term and g = (g0 , g1 , g2 ) is a vector of causal parameters. We assume that w is exogenous. This model will be referred to as model (1).
We have a large iid sample with n observations of y , d and w . Using this sample to regress y on a constant, d and w we obtain the OLS estimate ols = ╱0(ols) , 1(ols) , 2(ols)、\ .
1. Show that 1(ols) converges in probability to E(y|d = 1, w) ) E(y|d = 0, w). [4 marks]
2. Let us assume, for this question only that a family receives an income subsidy if and only
if its monthly income is below 2,000 Pounds. Explain why the exogeneity assumption
for w may then hold in model (1). If so, what does this imply for g~1ols ? [4 marks]
3. Let us now assume that to receive the subsidy a household must have a monthly income below 2,000 Pounds and they must send an application form.
(a) Explain why E(t|1, d, w) may no longer be equal to 0. [3 marks]
(b) Define z = ī{w < 2000}. Write down the conditions required for z to be a valid instrument for d in model (1). [3 marks]
(c) Derive the 2SLS estimator 2sls of g using (1, w, z) as an instrument for (1, w, d) in model (1). Specify all the data matrices that you use in your derivations. [4 marks]
(d) If the conditions given in item 3b are valid, show that 2sls is a consistent estimator of g . Does g2sls have a smaller variance than the IV estimator? [4 marks]
(e) The 2SLS estimation gives 1(2)sls = 12% and the estimated standard error for 1(2)sls is 4%. Test whether the income subsidy has a positive returns on test scores at the 5% significance level. [3 marks]
Part C [25 marks]
We have a sample {yit, X t} of N = 417 wines bottled from 1980 to 1995, where yit is the log price for wine i during the year s 4 {2000, 2005, 2010, 2015}, and
Xit = (aoОmait, flauОoit, insenit)\
is a vector of covariates measuring aromatic -aoОmait -, gustatory -flauОoit-, and color inten- sity -insenit- characteristics. We posit the linear specification:
yit = Xtg + ni + 5it , with 匝(5it|Xis, ni) = 0 for all s, r
where ni is the terroir of wine i, and 5it is an idiosyncratic disturbance term. Define
╱ yi1、 ╱ X1 、
Tyik1 = yi..2 and T k(X)i3 = X..2
(yiT . (XT .
Assume that {yi, Xi} is an i.i.d. sample and 5it and 5is are independent for s r.
1. Derive the First-Difference (FD) Estimator gˆfd . [5 marks]
2. Show that the FD estimator is consistent. State clearly all the assumptions/theorems that you use in your proof. [5 marks]
3. A wine expert argues that aromatic and gustatory characteristics of a wine are both related to its price. Write down a test based on the FD estimator (null and alternative hypotheses, test statistic and its asymptotic distribution under the null, critical values, and the decision rule) that could be employed to confirm the wine expert’s argument. [5 marks]
4. Can yit and yis, where s r, be correlated in this model? If yes, describe what the sources of this correlation can be. [5 marks]
Consider now the following specification with an individual linear time trend
yit = ai + uis + X t7 + (it , where 匝((it|Xi , ai, ui) = 0.
In this specification, each individual i is allowed to have its own linear time trend s ,与 uis. The individual linear time trend is an additional source of heterogeneity to the individual effect ai .
5. Does the strict exogeneity restriction 匝((it|Xi , ai, ui) = 0 allow for ai and ui to be cor- related? Does it allow for Xit+1 to be correlated with (it? Does it allow for Xit+1 to be correlated with ui? Justify. [5 marks]
Part D [25 marks]
We have an i.i.d. sample {yi, z} with yi a binary variable and zi a column vector of con- tinuous covariates. We specify the conditional probability of yi given zi as:
P (yi = 1|zi) = 1 ) exp[) exp(z go)] . (4.1)
For any number a, define Γ(a) := exp[) exp(a)]. Using this notation, one can write (4.1) as P (yi = 1|zi) = 1 ) Γ(z go) . We have:
= exp[) exp(z go)][) exp(z go)]go and
1. Let zki denote the k-th component of zi . Find the marginal effect of a change in zki on the conditional expectation of yi given zi . [5 marks]
2. Write down the log likelihood function and derive the first order conditions characterising the maximum likelihood estimator. [10 marks]
3. Explain the differences between the coefficient go, the maximum likelihood estimator of go and the maximum likelihood estimate of go . [5 marks]
4. Let gn denote the maximum likelihood estimator of go . Use the information matrix equal- ity to derive an estimator for the variance of gn . [5 marks]
2022-08-12