ST2023-EC220 - detailed solutions

EC Introduction to Econometrics Exam Solutions Summer Section A (Answer all questions. This section carries / of the overall mark.) Question [ . marks] Mining is dangerous. We are interested in understanding the relationship between the price of min- erals and the number of mining accidents. There are two hypothesized mechanisms whereby an increase in the price of a mineral may affect mining safety: ( ) safety may improve because mines have more revenue and are able to put more funds into safety, ( ) the opportunity cost of safety is higher, and mines may use more dangerous mining practices because they want to maximize pro- duction when prices are high. We are curious about which effect is dominant. We gather data from N mines. We have data on y i , the number of accidents in the mine, price i , the local market price of the mineral that is mined, and labour i , the number of labourers hired by the mine. We consider a regression model ln( y i ) = β 0 + β 1 ln( price i ) + β 2 labour i + β 3 labour 2 i + u i ( . ) After estimating regression ( . ) with OLS, we obtain estimates of coef cients, shown in the Stata output below. (a) Perform a t-test with the null hypothesis that the approximate elasticity of accidents with re- spect to price is , holding xed labour and labour squared. [ . marks] Solution : Because price i and y i are both in logs, β 1 is approximately interpreted as the elas- ticity of accidents with respect to price holding xed labour and labour squared. Thus we are performing a t-test with H 0 : β 1 = 1 . The t-statistic is: 0 . 5751439 - 1 0 . 3267842 = - 1 . 300 © LSE ST /EC R Page of 6

| - 1 . 300 | < 1 . 96 We fail to reject H 0 at the % signi cance level. (b) At present, the regression equation allows for a quadratic relationship between labour i and ln( y i ) . We want to change this. How could we use dummy variables to write a regression equation that allows for an effect of labour i on ln( y i ) that differs in ranges: ( ) labour i tak- ing values from to , ( ) labour i taking values to , and ( ) labour i taking values or greater? (In case it is not clear, the regression you write should allow for the effect of labour i on ln( y i ) to change depending on which category labour i is in, but, within a category, changing labour i doesn’t affect y i . E.g. labour i being rather than should have an effect, but labour i being rather than should not have an effect). [6 marks] Solution : Let us de ne dummy variables, x 1 i = 1[ labour i ≤ 99] , x 2 i = 1[100 ≤ labour i ≤ 999] , and x 3 i = 1[1000 ≤ labour i ] which are dummy variables that labour i takes a value in the range shown in brackets. One possible regression equation is: ln( y i ) = β 0 + β 1 ln( price i ) + β 2 x 1 i + β 3 x 2 i + u i Alternatively, one of x 1 or x 2 could be excluded and x 3 could be included. Or instead, the con- stant could be excluded and x 3 included. For the rest of the exam, ignore any change to the regression equation you may have written in (b). (c) Our data on price is the local market price at the time accidents are measured. Mines often have contracts for prices that are signed many months in advance. Thus, the market price recorded in our data may not be the price the mine receives. Describe how this could affect estimates. Regardless of whether you believe this will cause bias, also describe any other po- tential source of bias that may be a concern in regression ( ). Discuss the direction of the bias and the intuition of each. [8 marks] Solution : We observe ln( price i ) = ln( price i ) * + w i where ln( price i ) * is the true natural log of the price received by the mine and w i is measurement error. Classical measurement error is when w i is uncorrelated with all regressors and with the regression error. In this case, we have attenuation bias, and our estimate is biased toward , | b β 1 | < β 1 . Because our estimate is positive, the bias is negative. If the measurement error is not classical, then measurement error will in general cause bias, however, the direction of the bias will depend on the form of the measurement error. Are there any other sources of bias to be concerned about? Reverse causality is probably not a concern. The number of accidents in a single mine should not be causing the market price of a mineral to change. A student could conceivably argue that accidents reduce production, lower supply, and affect the market price. This is likely a minimal effect, but would cause pos- itive bias because the outcome increasing (accidents going up) is associated with lower sup- ply, and thus higher prices (treatment going up). Credit is given if the discussion is logical, but students do not need to claim there is bias due to reverse causality. There may be other con- founders. For example, the presence of toxic fumes or the use of explosive equipment makes a mine less safe, and causes the number of accidents to go up. Mines that use such explosive equipment might also be mining in a region where the geology requires this, and the miner- © LSE ST /EC R Page of 6

in the regression of interest. “Exclusion” – the only channel whereby averageother i affects y i is through ln(( price i ) . This might fail if other regions having higher (or lower) prices causes a mine to more aggressively produce, leading to more accidents. This seems unlikely, but should be considered. “As good as randomly assigned” – averageother i is uncorrelated with all unobserved deter- minants of y i . The concern regarding this was described in the answer to (d) regarding safety (use of explosives) when mining certain minerals. Students should describe potential concerns with each subassumption, but do not need to describe these speci c potential concerns. Exogeneity cannot be tested because the error term, u i , is not observable. © LSE ST /EC R Page of 6

Section B (Answer all questions. This section carries / of the overall mark.) Question [ . marks] (a) Consider a multiple regression model with k + 1 regressors (including intercept): MLR. The population model is y = β 0 + β 1 x 1 + ... + β k x k + u . MLR. We have a random sample, { ( y i , x i 1 , ..., x ik ) : i = 1 , ..., n } , from the model. MLR. We have no perfect collinearity among any of the regressors. MLR.Wild The regressors are “unrelated” to the error term (in a vague unspeci ed sense). MLR. The error term u is such that V ar ( u | x 1 , ..., x k ) = V ar ( u ) = σ 2 > 0 . (i) State the rst-order conditions for deriving ordinary least squares (OLS) estimators of the β parameters. Hence, or otherwise, compare and contrast the OLS approach for estima- tion of the β parameters with the method of moments (MM) approach. [ marks] Solution : (rescale marks to ) n X i =1 ( y i - ˆ β 0 - ˆ β 1 x i 1 - · · · - ˆ β k x ik ) = 0 n X i =1 x i 1 ( y i - ˆ β 0 - ˆ β 1 x i 1 - · · · - ˆ β k x ik ) = 0 . . . n X i =1 x ik ( y i - ˆ β 0 - ˆ β 1 x i 1 - · · · - ˆ β k x ik ) = 0 ( mark) For OLS, we choose ˆ β 0 , ˆ β 1 , . . . , ˆ β k to minimise n X i =1 ( y i - b 0 - b 1 x i 1 - · · · - b k x ik ) 2 with respect to b 0 , b 1 , . . . , b k . ( mark) On the other hand, in the MM approach, (i) we make assumptions about population mo- ments (this yields equations involving our unknown parameters); (ii) we estimate popula- tion moments with their sample counterparts; (iii) we then solve the equations to obtain estimators for our parameters. ( mark) Within the given regression setting, the MM approach yields the same set of k + 1 mo- ment equations as the system of k + 1 FOC conditions from the LS approach. Hence the estimators will be equivalent. ( mark) (ii) Explain how the OLS estimator of β 1 , say ˆ β 1 , can be obtained from a simple regression of y on some suitably-constructed univariate regressor rather than a multiple regression of y on an intercept and the full set of regressors x 1 , x 2 , ..., x k . [ marks] © LSE ST /EC R Page of 6

( marks) Step : Decompose our estimator as follows • So we get ˆ β 1 = β 1 SST x + ∑ n i =1 ( x i - ¯ x ) u i SST x = β 1 + ∑ n i =1 ( x i - ¯ x ) u i SST x • Now de ne w i = ( x i - ¯ x ) SST x • Then we have ˆ β 1 = β 1 + n X i =1 w i u i • Sampling error ˆ β 1 - β 1 is written by a linear function of u i ’s • w i ’s are all functions of X = { x 1 , . . . , x n } ( mark) (iv) Propose the weakest form of MLR.Wild that would be needed to establish consistency of ˆ β 1 for β 1 , and call it MLR.Weak . Prove consistency under MLR.Weak. [ marks] Solution : (rescale marks from 6 to ) We drop the second subscript on x i 1 since there is only one non-constant regressor. Our assumption MLR.Weak is: (i) E ( u ) = 0 , and (ii) E ( xu ) = 0 . ( marks) OLS estimator is written as: ˆ β 1 = β 1 + n - 1 ∑ n i =1 ( x i - ¯ x ) u i n - 1 ∑ n i =1 ( x i - ¯ x ) 2 Look at numerator of sampling error term: n - 1 n X i =1 ( x i - ¯ x ) u i = 1 n n X i =1 x i u i - ¯ x ¯ u By LLN, plim (¯ x ) = E ( x ) plim (¯ u ) = E ( u ) = 0 plim 1 n n X i =1 x i u i ! = E ( xu ) = 0 ( marks) By properties of the plim operator, plim ( ˆ β 1 ) = β 1 + plim ( 1 n ∑ n i =1 x i u i ) - plim (¯ x ¯ u ) plim ( 1 n ∑ n i =1 ( x i - ¯ x ) 2 ) = β 1 + E ( xu ) - E ( x ) E ( u ) V ar ( x ) = β 1 © LSE ST /EC R Page of 6

( marks) (v) Propose how to strengthen MLR.Weak to establish unbiasedness of ˆ β 1 for β 1 , and call the new assumption MLR.Strong . Prove unbiasedness under MLR.Strong. [ marks] Solution : (rescale marks from 6 to ) We drop the second subscript on x i 1 since there is only one non-constant regressor. Our assumption MLR.Strong is: E ( u i | x i ) = 0 . Together with MLR. , this will suf ce for unbiasedness. ( marks) • We have E ( ˆ β 1 | X ) = E β 1 + n X i =1 w i u i X ! = β 1 + n X i =1 E ( w i u i | X ) = β 1 where we use linearity of the expectations operator ( marks) • Thus, for any X , we get the same conditional expectation E ( ˆ β 1 | X ) = β 1 and the law of iterated expectations implies E ( ˆ β 1 ) = β 1 ( marks) (b) Suppose we have an “original” tted regression given by log Y i = b 1 + b 2 log X i + e i , where b 1 and b 2 are coef cients estimated via OLS, and e i is the OLS residual, for i = 1 , ..., n . Now suppose units of measurement change so that X * i = λX i , where λ is some non-zero nite constant, and that log Y i is regressed on and log X * i . Our “new” tted regression is log Y i = b * 1 + b * 2 log X * i + e * i , where b * 1 and b * 2 are new coef cients estimated via OLS, and e * i is the new OLS residual. (i) Derive the relationship between the original and new coef cients estimated via OLS.( Hint: Start by showing that log X * i - log X * = log X i - log X, where a bar denotes the sample mean.) [ marks] © LSE ST /EC R Page 8 of 6

Question [ . marks] (a) Suppose that you are hired as class teacher for an undergraduate econometrics course and that your students ask you the questions listed below. Please provide suitable answers. (i) “Can you provide a real (or hypothetical) example of heteroscedasticity?” [ marks] Solution : (rescale marks from to )Any sensible example that reveals an understand- ing of heteroscedasticity should earn full marks. It can be algebra and words, or a dia- gram and words, or even just well-explained words alone. (ii) “What is the Gauss-Markov theorem? Why is it useful?” [ marks] Solution : (rescale marks from to )Using the language of Question (a), under MLR. – MLR. , MLR.Strong and MLR. , ... ( mark) the OLS estimator is the Best Linear Unbiased Estimator (BLUE). ( mark) If we take any linear unbiased estimator ˜ β 1 , then it always holds V ar ( ˆ β 1 | X ) ≤ V ar ( ˜ β 1 | X ) ( mark) Implication: If we are interested in linear unbiased estimators, then we need look no fur- ther than OLS ( mark) (iii) “Say I estimate a simple regression model via OLS. How do I prove that tted values of the dependent variable are uncorrelated with the OLS residuals?” [ marks] Solution : (rescale marks from to ) The numerator of the sample correlation coef - cient for ˆ Y and e can be decomposed as follows, using the fact that ¯ e = 0 : 1 n n X i =1 ˆ Y i - ˆ Y ( e i - ¯ e ) = 1 n n X i =1 ( [ b 1 + b 2 X i ] - b 1 + b 2 ¯ X ) e i = 1 n b 2 n X i =1 ( X i - ¯ X ) e i = 0 by OLS rst-order conditions. Hence the correlation is zero. (iv) “I want to test for joint signi cance of all regressors excluding the intercept in a multiple regression model. What assumptions will I need? What test statistic shall I use? What is the distribution of the test statistic under the null? What rejection rule should I use?” [ . marks] Solution : (rescale marks from to . ) Using the language of Wooldridge (and lecture notes), we will need assumptions MLR (linearity), MLR (random sampling), MLR (no perfect collinearity), MLR (strict exogeneity), MLR (homoscedasticity), and MLR6 (nor- mality). ( mark) © LSE ST /EC R Page of 6

The test statistic is V = ( SSR 0 - SSR 1 ) /k SSR 1 / ( n - k - 1) , where n is the number of observations, SSR 0 and SSR 1 are the restricted and unre- stricted residual sums of squares respectively, and k is the number of non-intercept re- gressors. ( marks) The distribution under the null is F ( k, n - k - 1) . ( mark) We reject if and only if V > V (1 - α ) , where V (1 - α ) is the 100(1 - α ) -th percentile on the CDF of the F ( k, n - k - 1) distribution, and α ∈ (0 , 1) is a given signi cance level. ( mark) (b) Let us consider the following system of simultaneous equations y 1 = α 1 y 2 + α 2 x 1 + α 3 x 2 + u 1 ( . ) y 2 = β 1 y 1 + β 2 x 1 + β 3 x 3 + β 4 x 4 + u 2 , ( . ) where ( u 1 , u 2 ) are i.i.d. errors with zero mean, V ar ( u 1 ) = σ 2 1 , V ar ( u 2 ) = σ 2 2 , and Cov ( u 1 , u 2 ) = σ 12 . The endogenous variables are ( y 1 , y 2 ) and ( x 1 , x 2 , x 3 , x 4 ) are exogenous variables. (i) Explain the simultaneity issue associated with the above simultaneous equation model (SEM) intuitively (no derivations expected). Outline a real-world scenario for which the given model may be useful. [ marks] Solution : (rescale marks from to ) The problem here is reverse causality: both equa- tions have their own ceteris paribus interpretation, the simultaneity gives rise to a corre- lation between the two endogenous variables that are jointly determined in this system which makes it that when we estimate these structural equation in isolation by OLS we get inconsistency. One example (discussed) is police&crime. (ii) You are told that in the above SEM there exists a perfect linear relation between x 2 and x 3 , in particular, x 3 = 2 x 2 . Obtain the reduced form equations for y 1 and y 2 , recognising this exact linear relation between x 2 and x 3 . [ marks] Solution : (rescale marks from 6 to ) Let us rst rewrite the SEM accounting for this perfect linear relation: y 1 = α 1 y 2 + α 2 x 1 + α 3 x 2 + u 1 ( . ) y 2 = β 1 y 1 + β 2 x 1 + 2 β 3 x 2 + β 4 x 4 + u 2 , ( . ) Substitute ( . ) into ( . ) we have: y 1 = α 1 ( β 1 y 1 + β 2 x 1 + 2 β 3 x 2 + β 4 x 4 + u 2 ) + α 2 x 1 + α 3 x 2 + u 1 (1 - α 1 β 1 ) y 1 = ( α 1 β 2 + α 2 ) x 1 + ( α 3 + 2 α 1 β 3 ) x 2 + α 1 β 4 x 4 + u 1 + α 1 u 2 y 1 = ( α 1 β 2 + α 2 ) x 1 + ( α 3 + 2 α 1 β 3 ) x 2 + α 1 β 4 x 4 + u 1 + α 1 u 2 (1 - α 1 β 1 ) , © LSE ST /EC R Page of 6

Question [ marks] Consider the following linear regression model y t = β 0 + ρy t - 1 + β 1 z t - 1 + β 2 z t - 2 + u t with | ρ | < 1 t = 1 , ..., n ( . ) where { y t } and { z t } are stationary and weakly dependent processes. We have a sample of n ob- servations we can use to estimate this equation which we label t = 1 , ..., n for convenience (this means we assume that y 0 , z 0 , z - 1 are available). There is no perfect multicollinearity in the regres- sors. (a) Stationarity and weak dependence are important concepts that will allow us to apply the usual laws of large numbers and CLT. Explain what the concepts of stationarity and weak depen- dence mean. [ marks] Solution : (rescale marks from to )Text book discussion of stationarity and weak depen- dence (happy to accept students discussing the concept of covariance stationarity as long at it is clearly indicated) ( mark each) . The importance of these concepts are that they ensure that we can apply the usual laws of large numbers and CLT, that will allows us to use asymptotic inference as in the cross sec- tional setting (no need for normality, or strict exogeneity which is important in time series analysis. ( marks) (b) What (reasonable) assumptions do you need to make about the errors u t to ensure that OLS on equation ( . ) provides consistent estimators of the parameters ( β 0 , β 1 , β 2 , ρ ) ? Provide a clear explanation of the assumptions you make. [ marks] Solution : (rescale marks from to ) To ensure consistency, we need to assume contempo- raneous exogeneity : E ( u t | y t - 1 , z t - 1 , z t - 2 ) = 0 ( marks) This assumption will ensure Cov ( u t , y t - 1 ) = 0 , Cov ( u t , z t - 1 ) = 0 and Cov ( u t , z t - 2 ) = 0 needed for consistency of OLS (can show using LIE), so that the errors and the regressors (the way they appear in the regression) are uncorrelated. ( mark) These assumptions are weaker than strict exogeneity, an assumption that we cannot have given that we have y t - 1 as regressor anyway. This restriction does permit, e.g., Cov ( u t , z t +1 ) 6 = 0 , so that future regressors may respond to shocks which is important in time series analysis. ( mark) Given the assumptions on the stochastic processes { y t } and { z t } , u t will be stationary and weakly dependent automatically. (c) Discuss what the short run and long run effect of z on y are in terms of the parameters and what they tell you. Under the assumptions given in (b), discuss how you can obtain a con- sistent estimator of the long run impact. Prove your claim. In your answer you do not have to prove the consistency of the OLS estimator of ( β 0 , β 1 , β 2 , ρ ) . [ marks] Solution : (rescale marks from 6 to ) The short run effect of z on y here is zero as z t does not appear as a regressor, that is there is no instantaneous effect of z changing ( marks) . © LSE ST /EC R Page of 6

To obtain the long run effect (requires | ρ | < 1 ), let us de ne y e and z e as the equilibrium value so that y * = β 0 + ρy * + β 1 z * + β 2 z * The error term is taken to be its mean value zero in the long-run. Upon rewriting, we obtain: y * = β 0 1 - ρ + β 1 + β 2 1 - ρ z * The long run effect is β 1 + β 2 1 - ρ . It shows the effect a permanent unit change in z t has on y t in the long run (after all lagged effects have been incorporated). ( marks) . To obtain a consistent estimator, we plug in the OLS estimator. Using the properties of the plim operator, we observe: plim ˆ β 1 + ˆ β 2 1 - ˆ ρ ! = plim ˆ β 1 + plim ˆ β 2 1 - plim ˆ ρ = β 1 + β 2 1 - ρ establishing its consistency. ( marks) . (d) You want to test the hypotheses that u t in ( . ) exhibits serial correlation. Discuss how you can test the hypothesis that there is no autocorrelation against the alternative that u t follows a stationary, weakly dependent AR( ) process, i.e., u t = φu t - 1 + v t , | φ | < 1 ( . ) where v t is iid ( 0 , σ 2 v ) error that is independent of y t - 1 , y t - 2 , .. and z s for all s. [ marks] Solution : (rescale marks from to ) You want to test H 0 : φ = 0 H 1 : φ 6 = 0 To conduct this test, we rst obtain the OLS residuals: ˆ u t = y t - ˆ β 0 - ˆ ρy t - 1 ˆ β 1 z t - 1 - ˆ β 2 z t - 2 Next we estimate the following regression: ˆ u t = γ 0 + ρ ˆ u t - 1 + γ 1 y t - 1 + γ 2 z t - 1 + γ 3 z t - 2 + e t Under TS. ’-TS. ’, our asymptotic t test is, t = ˆ ρ se (ˆ ρ ) a ∼ N (0 , 1) under H 0 (we should assume TS. ’ as well if we do not use white robust standard errors). At the % level of signi cance we should reject if the realisation of our test statistic is larger (in abso- lute value) than . 6, or equivalently if its p-value is smaller than %. (e) Say your test in (d) does not nd evidence of autocorrelation. Discuss how you can test whether β 1 + β 2 = 0 . 1 against the alternative that β 1 + β 2 > 0 . 1 . In your answer discuss what regres- sion you should estimate if you want to be able to obtain the test statistic directly using your regression output. [ marks] © LSE ST /EC R Page of 6

direct effect on y t (exclusion). Let us lag ( . ) one period: y t - 1 = β 0 + ρy t - 2 + β 1 z t - 2 + β 2 z t - 3 + u t - 1 This shows that z t - 3 can be used as an instrument for y t - 1 . In fact, we can also propose to use y t - 2 here as an instrument as the MA( ) dependence structure ensures that y t - 2 will not exhibit any correlation with u t (exogeneity). ( marks) We can proceed using SLS (or IV), where ( marks) • Step : Obtain the tted values for y t - 1 from running the following regression: y t - 1 = π 0 + π 1 z t - 1 + π 2 z t - 2 + π 3 z t - 3 + e t The tted values ˆ y t - 1 = ˆ π 0 + ˆ π 1 z t - 1 + ˆ π 2 z t - 2 + ˆ π 3 z t - 3 • Step : Run the following regression y t = β 0 + ρ ˆ y t - 1 + β 1 z t - 1 + β 2 z t - 2 + v t END OF PAPER © LSE ST /EC R Page 6 of 6