Exercises Econometric Models. Let u t be a scalar random variable such that E(u t j I t ) =, t = ; ; ::::, where I t is the (stochastic) information set available at time t. Show that under the hyothesis E(u t j I t ) = " = const, the sequence u t is White Noise. Solution: We check that (a) E(u t ) =, (b) E(u t ) = " and (c) E(u t u t j ) =, j = ; ; :::: First, E(u t ) = E [E(u t j I t )] = E() = by the law of iterated exectations, hence condition (a) is ful lled. Second, E(u t ) = E E(u t j I t ) = E( ") = " hence (b) is ful lled. Finally, in (c), E(u t u t j ) = E[E(u t u t j j I t )] for j = ; ::::Since u t j I t, one has E(u t u t j ) = E[E(u t j I t )u t j ] = E(u t j ) = for j = ;,..., hence also condition (c) holds.. Let fx t g tn be a martingale di erence sequence (with resect to its own ast). Which of the following sequences necessarily are martingale di erences (with resect to their own ast)? Justify from the de nition. (a) Z t = X t. his is an m.d.s. because EjZ t j = EjX t j < and E(Z t jz t ; :::; Z ) = E(X t jx t ; :::; X ) = E(X t jx t ; :::; X ) = E(X t jx t ; :::; X ) = ; using that fx ; :::; X t g and fx ; :::; X t g contain the same information and X t is an m.d.s. (b) W t = X t. Also this is an m.d.s because EjW t j = EjX t j < and E(W t jw t ; :::; W ) = E(X t jx t ; :::; X ) = E(E(X t jx t ; X t ; :::; X )jx t ; :::; X ) = E(jX t ; :::; X ) = by the law of iterated exectations and because X t is an m.d.s. (c) U t = X [t=]+, where [] is the integer-art function. As the sequence in question is X ; X ; X ; X ; X ; :::; it holds that in general, so U t need not be an m.d.s. (d) V t = Xt +X t E(U ju ; U ) = E(X jx ; X ) = X 6= E( Xt jx +Xt t ; :::; X ): Now we have an m.d.s. again as EjV t j EjX t j < because X t is an m.d.s. and E(V t jv t ; :::; V ) = E(E(V t jx t ; :::; X )jv t ; :::; V ) Xt X t = E E + Xt E( + Xt jx t ; :::; X )jx t ; :::; X jv t ; :::; V X t X t = E E( + Xt jx t ; :::; X ) E( + Xt jx t ; :::; X )jx t ; :::; X ) jv t ; :::; V = E(jX t ; :::; X )jv t ; :::; V ) = using the law of iterated exectations.. Summarize what the term misseci cation analysis means.. De ne what a consistent estimator is.. onsider the model y t = + y t + z t + u t ; u t WN(; u); where it is known that E(y t u t ) = but E(z t u t ) 6=. Assuming ergodicity (and hence, the validity of an LLN), nd the robability limit of the OLS estimator of = ( ; ; ) : Under what condition is estimated consistently? Hint: use the usual decomosition and then... ^ = + ( X X) ( X u)
4. onsider the regression model y t = z t + u t, t = ; ; :::; where z t is a vector of contemoraneous exlanatory variables. Suose it is known that u t obeys u t = u t + " t, " t WN(; "), t = ; :::; where < < is a known coe cient and u is indeendent of f" t g and has E(u ) = ; E(u ) = " : A. Derive the covariance matrix :=E(uu ). B. Show that the model can be re-written as a articular regression model whose exlanatory variables include one lag of y t and one lag of z t, resectively (other than a White Noise disturbance). Solution: A. he matrix = E(uu ) is: E(u ) E(u u ) E(u ) = 6 4..... 5 : E(u u ) E(u u ) E(u ) In this matrix, the diagonal elements E(u t ), t = ; :::;, equal the variance of the AR() rocess u t = u t +" t, hence, E(u t ) = " for t = ; :::. Now let us focus on E(u t u t j ): this quantity is an autocovariance of an AR() rocess: ov(u t ; u t j ) = E(u t u t j ) = j ", j = ; ; ::::; : herefore, = " 6 4..... 5 : B. Relacing u t in the rst equation with u t = u t + " t yields y t = z t + u t + " t, t = ; ; :::; ; since u t = (y t z t ), one has so eventually y t = z t + (y t z t ) + " t, t = ; ; :::; ; y t = y t + z t z t + " t. It is clear that this model includes y t and z t among the regressors and a White Noise disturbance. 5. De ne the concets of a DGP and a statistical model. 6. onsider the regression model t = + Y t + t + Y t + u t, t = ; ; :::; () in which t = consumtion growth of a given country; Y t = disosable income growth of the country; u t m:d:s:(; u). Assuming stationarity and ergodicity of the involved variables, what are the roerties of the OLS estimator of the arameters in this setu?. onsider the dynamic regression model t = + Y t + t + Y t + u t, t = ; ; :::; () in which t = consumtion growth rate of a given country; Y t = disosable income growth rate of the country; u t W N(; u). A. Imagine that at time t =, < <, the arameter is otentially a ected by a structural break (a change) due to a scal reform. Re-write the model such that the ossibility of the structural break is taken into account.
B. Proose a test statistic for the null hyothesis of no structural change. Solution: A. Under the assumtion that at time t =, < < the arameter changes (due to the scal reform), the model can be re-written in the form t = + Y t + Y t D t + t + Y t + u t, t = ; ; :::; () in which D t is a dummy variable de ned by D t := t < t : It is seen that for t < the imact of disosable income growth on er-caita consumtion growth is given by, while for t the imact is + : B. he hyothesis of no change in corresonds to the null H : = : 8. [EXAM OF MAY ] onsider the model b t = + oil t + b t + u t, u t m:d:s:; (4) t = ; ; :::; ; where b t is the change of the rice of fuel in month t and oil t is the change of the rice of oil in month t. he OLS estimation of, and yields: ^ =.5, s.e.(^ )=., ^ =., s.e.(^ )=.5, ^ =.88, s.e.(^ )=., where heteroskedasticity-robust standard errors are reorted. Further, R = :45; LM (Breusch-Godfrey) test for autocorrelated disturbances of order 6 =. [-value =.89] LM (Breusch-Pagan) test for homoskedasticity of order 6 = 4.5 [-value =.] : You may assume that the variables involved are stationary. A. o the extent allowed by the rovided information, tell whether the model is correctly seci ed and if the OLS estimator can be alied meaningfully in this context. B. onsider the null hyothesis H : = : onduct a test of H against H : 6= and interret the result.. Imagine that in estimating the model above, the econometrician has ignored the following information relative to oil t : oil t = oil t + " t, j j<, " t W N(, ") (5) E(" t u t ) 6= ; what can be said about the roerties of the OLS estimator in this context? Which method can be alied to estimate the arameters of the model in Eq. (5) consistently? Solution A. If we x the tye I error of the tests to = :5 (5%), the estimated model does not show any sign of autocorrelated disturbances because the -value associated with the LM test for the absence of autocorrelation is greater than ; instead, it seems that there is some heteroskedasticity in the disturbances. he tests rovide no evidence against the consistency of the OLS estimator. herefore, OLS can be used meaningfully in conjunction with heteroskedasticity-consistent standard errors, although it should be noticed that OLS is not e cient because of the detected heteroskedasticity. B. A simle test-statistic for H is the "t-ratio" ^ s.e.(^ ) = :: If one comares the realized value of the test statistic,., with the critical value taken from the standard normal distribution (the t-ratio is asymtotically N(; ) under H ), the conclusion is
that there is no signi cant imact of oil rice changes on the rices of fuel. Note that the test above is a Wald-tye test.. he main consequence of neglecting the information in Eq. (5) is that the OLS estimator is no longer consistent for the arameters of eq. (4) due to an endogeneity bias. One ossible x to this roblem is to aly the IV estimator based e.g. on the vector of instruments: z t = (; oil t ; b t ) : 9. onsider the DGP y t = :5y t + :z t + u t, u t WN(,) z t = :y t + " t, " t WN(,:5) ov(u t ; " t ) = t = ; :::; (in which all variables are stationary and ergodic) and suose that the econometrician seci es the following model for y t : y t = y t + u t, u t WNN(, u) (6) t = ; :::; : What value does the OLS (ML) estimator of estimate consistently? (Recall that X X = P t= y t and X y = P t= y t y t ). SOLUION: he robability limit of the OLS (ML) estimator can be found as follows: ^ OLS = ^ ML = (X X) (X y) = P t= y t y t P t= y t = using the DGP = = P t= y t [:5y t + :z t + u t ] P t= y t P t= = :5 + : y t P t= y t P t= = :5 + : y P t [ :y t + " t ] t= P + y t t= y t P z t t= + y t u t u t P t= y t P t= y t P t= = :5 :6 + y P t " t t= + y t u t P t= y t P t= = :5 :6 + y t " t P + t= y t observe that due to the stationary and ergodicity: hence hus, we conclude that X y t " t! E(y t " t ) = t= X y t u t! E(y t u t ) = t= P t= y t P t= y t u t P t= y t X yt! E(yt ) = E(yt ) < ; t= ^ ML! :5 :6 + E(y t " t ) E(y t ) ^ ML! :5 :6 = :44: + E(y t u t ) E(yt : ) he value of.44 is di erent from.5 because y t in eq. (6) acts also as a roxy for the omitted variable z t.. onsider the regression model y t = x t + u t, u t WNN(; u), t = ; :::; () and the ossibility that at time < < a structural break might have changed all elements of (but not u). Formalize the null hyothesis H :absence of structural breaks and roose a test statistic that has a standard limit distribution under the null hyothesis. What is this limit distribution? ; 4
. Let q t be the quantity of cars (of brand XX) roduced in quarter t, t be the corresonding average rice at quarter t, and fuel t be the average rice of fuel at quarter t. onsider the dynamic regression model: q t = + t + fuel t + q t + u t, t = ; ; :::; in which E(u t j fuel t ; q t ) =, E(u t j fuel t ; q t ) = u, t = ; ; :::; and E( t u t ) 6=. Assume that all the involved variables are stationary and erodic. Show that the OLS estimator of = (,,, ) is not consistent for. Solution: Since E( t u t ) 6=, it holds that E(x t u t ) = E 6B 4@ t fuel t q t A u t 5 = note that use has been made of the law of iterated exectations, e.g. E(fuel t u t ) = E[E(fuel t u t j fuel t ; q t )] = E[fuel t E(u t j fuel t ; q t )] = E[fuel t ] =. Under the assumtion that Ex t x t is invertible, it follows that ^ OLS = + (Ex t x t) E(x t u t ) 6= because E(x t u t ) 6=. B @ 6=. In the context of the revious exercise, roose a consistent estimator of : Solution: One ossibility is to use the IV estimator obtained with the vector of instruments: z t = B t @ fuel t A : q t Indeed, while = E(z t u t ) = E 6B 4@ zx = E(z t x t) = E 6B 4@ 6 4 = E 6 4 t u t u t fuel t u t q t u t t fuel t q t A5 = B @ A ; A A (; t; fuel t ; q t ) t fuel t q t t t t t fuel t t q t fuel t fuel t t fuel t fuel t q t q t q t t q t fuel t q t E( t ) E(fuel t ) E(q t ) E( t ) E( t t ) E( t fuel t ) E( t q t ) E(fuel t ) E(fuel t t ) E(fuel t ) E(fuel t q t ) E(q t ) E(q t t ) E(q t fuel t ) E(q t ) where, due to the assumed stationarity, we have used the roerty E(q t ) = E(q t ), etc. his matrix must have rank 4 for the instruments to be relevant (aart from valid).. onsider the DGP (based on stationary ergodic variables): and an econometrician who estimates y t = + y t + z t + u t, u t m.d.s.(; u) y t = + y t + u t : Under what general condition is the OLS estimator of = ( ; ; ) consistent for? Solution: It is convenient to re-write both models in matrix form. First, consider the DGP: 5 5 5 y = X + Z + u 5
where 6 y:= 4 y. y 6 5, X:= 4. 6 Z:= 4 z. z y. y 6 5, u:= 4 hen consider the model estimated by the econometrician: 5, := u. u 5 : y = X + u : he OLS estimator of comuted by the econometrician is given by: ^:=(X X) X y: If we relace y with the true model (DGP), the OLS estimator reads: ^:=(X X) X [X + Z + u] = + (X X) (X Z) + (X X) (X u) = + ( X X) ( X Z) + ( X X) ( X u): With stationary ergodic variables we have: ( X X) B! E @ (; y y t ) A = xx t x t x t ( B X u)! E @ u y t A = t so that ( X Z)! B E @ y t x t x t z t A = xz ^! + xx xz + xx = + xx xz : he last formula shows that xz = (which slits into E(x t z t ) = xz = or = ) is a necessary and su cinet condition for the consisteny roery ^!. 4. onsider the model in which it is known that and y t = + y t + g t + u t, u t WN(; u) g t = + g t + g t + t, t WN(; ) ov(u t ; t ) 6= : Show that it is not ossible to estimate = ( ; ; ) consistently by OLS. Proose an IV estimator of = ( ; ; ) and in articular indicate which are the r chosen instruments z t. Show that given the chosen instruments, it holds that E(z t u t ) = r and nally derive the matrix zx = E(z t x t) (x t = (; y t ; g t ) ). 5. De ne what endogeneity bias means and rovide an examle of a model a ected by this issue. 6. De ne the IV estimator and rove its consistency under aroriate conditions. 6
. Which result do we use when we rove the asymtotic normality of the OLS estimator? Solution: We use a entral Limit heorem for martingale di erence sequences, which asserts that... 8. Show that the IV estimator is a secial case of the OLS estimator. 9. Show that the OLS estimator is a secial MM estimator.. [FROM HE EXAM OF JUNE ] onsider the dynamic linear regression model in which it is known that and y t = + z t + y t + u t, t = ; ; :::; (8) z t = z t + " t, - < < ut ~m:d:s: " t ; u " : What are the roerties of the OLS estimator of :=( ; ; ) in this context? Solution. It can be noticed that the dynamic linear regression model is seci ed such that the regressors in the vector x t :=(z t,y t ) are uncorrelated with the disturbance term u t (note indeed, that z t and u t are uncorrelated because ov(u t ; " t ) = ). Accordingly, the OLS estimator of :=( ; ; ) is (a) consistent for and (b) asymtotically Gaussian, if the variables involved are stationary and ergodic (this requires j j < ).. [FROM HE EXAM OF JUNE ] onsider the model y t = + z t + y t + u t, u t m:d:s:(, u), t = ; ; :::; 5; (9) suose that all variables are stationary and ergodic and that the OLS estimation of, and has roduced: ^ :=.5, s.e.(^ ):=. ^ :=., s.e.(^ ):=. R = :45 LM (Breusch-Godfrey) test for autocorrelated disturbances u to lag order = 8. [-value =.6] LM (Breusch-Pagan) test for homoskedasticity = 5.5 [-value =.] ; where the standard errors are heteroskedasticity-consistent. ell whether z t exlains y t. Discuss how the LM (Breusch-Godfrey) test statistic for autocorrelated disturbances reorted above has been obtained (i.e. tell what Gretl does to comute the test statistic). Solution ell whether z t exlains y t is another way to say: is the regressor z t signi cant in the above regression model?, which in turn is equivalent to the question is signi cant?. In other words, we have to test H : := vs H : 6=. o address this question we use: t := ^ s:e:(^ ) :=.. = ; if the signi cance level of the test is re- xed at the 5%, we know that the asymtotic critical values of the test are.96 (it is a two-sided test!). We observe that the realized value of the test statistic, equal to, lies within the (95%) accetance region (.96, +.96), hence we do not reject the null H : :=. o obtain the LM (Breusch-Godfrey) test statistic for autocorrelated disturbances (u to lag order ) we can consider the following stes:. estimate the original model by OLS and take the residuals ^u t :=(y t ^ +^ z t +^ y t ); t = ; ; :::; ;. consider the auxiliary regression: ^u t = + z t + y t + ^u t + " t, t = ; ; :::;
and estimate this model by OLS;. given the auxiliary regression, the LM (Breusch-Godfrey) test for autocorrelated disturbances (u to lag order ) corresonds to a test for H : := against H 6=. In articular, GREL rovides R as a test statistic for H against H, where R refers to the auxiliary regression, not to the original one.. onsider the linear regression model y t = + z t + y t + u t, u t = u t + " t, -<<; " t WNN(; "); () Exlain why the OLS estimator of =( ; ; ) is not consistent for if 6=. Proose a test for the null hyothesis of uncorrelated disturbances.. [FROM HE EXAM OF JUNE ] De ne the tye-i error and -value of a statistical test for a hyothesis H against H. De ne the concet of asymtotic size of a test for H against H : Solution Given a samle of length and the test statistic S for H against H and the decision rule: the tye-i error of the test is de ned as reject H if s cv ; (accet H if s < cv ; ), := Pr(reject H j H ):=Pr (S cv ; ) where cv ; is the ( ) ercentile of the distribution of S under H :Given the realized value of the test statistic on the observed samle S :=s, the -value of the test is de ned by -value = Pr(S s j H ): he concet of asymtotic sise is related to testing hyotheses in large samles. Given the null distribution of S for large (! ), the asymtotic size of the test is given by :=lim! Pr(S cv ; j H ) where the value of (nominal signi cance level) is re- xed a riori. 4. [FROM HE EXAM OF JUNE ] onsider the dynamic linear regression model in which it is known that and with u;" 6=. y t = + z t + y t + u t, t = ; ; :::; () z t = " t ut ~WNN " t ; u u;" u;" ". What are the asymtotic roerties of the OLS estimator of :=( ; ; ) in this context?. Suose that the econometrician has the susicion that a break in the arameters and might have occurred at time B (< B < ); re-write the model such that the break is taken into account and roose a test for the null that no structural break occurred.. Describe how a LR test for the null in item () might be also comuted. SOLUION.. It is immediately seen that all the regressors enter the model with a lag, hence x t :=(; z t ; y t ) is redeterminaed by construction and the OLS estimator of :=( ; ; ) will be consistent and asymtotically Gaussian if all the variables are stationary and ergodic (this requires j j < to hold).. If a structural break may a ect the arameters and at time < B <, the model can be re-written in the form y t = + D t + z t + D t z t + y t + u t, where D t := t B elsewhere : 8
he hyothesis that no break occurs against the alternative of a break corresonds to against H : = and = H : 6= or 6= : A test of H vs. H can be based on the Wald test statistic: where W := (R^ r) n^ ur^ xx R o (R^ r) R:= he test statistic satis es under H : since there are q:= restrictions under test., := B @ W! D () A, r:=. An LR test might be conducted by estimating the restricted model (under H ) and the unrestricted (under H ) model and then comuting the test statistic which under H satis es similarly to the Wald statistic. y t = + z t + y t + u t y t = + D t + z t + D t z t + y t + u t LR := [log L(^) H LR! D () 5. [FROM HE EXAM OF JUNE ] onsider the model log L(^) ] unrestricted (H ) y t = + z t + y t + u t, t = ; ; :::; ; () where it is known that z t = y t + z t + " t ; ut ~WNN ; u " t " and the initial values y ; z are indeendent of (u t ; " t ). Under what condition is the OLS estimator of :=( ; ; ) consistent for? If this condition is violated, discuss how a consistent estimator for :=( ; ; ) could be obtained. Assume throughout that the true arameter values are such that (y t ; z t ) is a stationary rocess. SOLUION It holds that E(z t u t ) = E [( y t + z t + " t )u t ] = E(y t u t ) = E[( + z t + y t + u t )u t ] = E(z t u t ) + u: herefore, = is a necessary and su cient condition for E(z t u t ) = to hold (in words, this is the condition that z t is not a ected by y t ). Under this condition the OLS estimator is consistent for, otherwise an endogeneity roblem is resent (z t is endogenous!). A ossible x to this roblem could be to estimate :=( ; ; ) by IV using, e.g., the vector w t = (; z t ; y t ) as a vector of instruments. his choice of instruments satis es the rank condition for identi cation i Efwt(; z t ; y t )g is a matrix of rank, which for some arameter values is not the case (check the rank for = = = =, 6= ). For the latter arameter combination z t cannot be instrumented by variables among fy s ; z s : s = ; :::; t g and the information set needs to be enlarged by other variables using knowledge of the nature of the henomenon under study. : 9
6. [FROM HE EXAM OF JULY ] onsider the dynamic linear regression model where it is known that and y t = + z t + y t + u t, t = ; ; :::; ; () z t = z t + " t, - < < ut ~WNN " t ; u u;" u;" " with u;" 6=. Proose a consistent estimator of :=( ; ; ). Solution. Simle algebra shows that in this model z t and u t are correlated, hence the OLS estimator of :=( ; ; ) is not consistent for. A consistent estimator of is the IV variable estimator with an aroriate choice of instruments; for instance, g t :=(; z t ; y t ) could be used rovided that 6=. For = it holds that Ez t z t = and the rank condition for identi cation is not satis ed by g t. In fact, for = it is not ossible to instrument z t with variables among fy s ; z s : s = ; :::; t g and the information set needs to be enlarged by other variables using knowledge of the nature of the henomenon under study. :. [FROM HE EXAM OF JULY ] onsider the linear regression model y t = + z t + y t + u t, t = ; ; :::; : (4) Describe the Godfrey and Breush diagnostic test for autocorrelated disturbances (exlain which the null hyothesis is and how the test works in ractice). Answer. For a given m, consider the autoregression he null and alternative hyotheses are u t = u t + ::: + m u t m + " t ; " t m:d:s:(, "): H : =, =,..., m =, H : at least one i 6=, i = ; :::; m: he test of H can be conducted as follows. Let ^u t be residuals obtained from the estimation of Eq. (4). Estimate the auxiliary regression ^u t = + z t + y t + ^u t + ::: + m^u t m + error t and test for = ::: = m = above using as test statistics R or a Wald-tye statistic. he test statistic will have (m) asymtotic distribution under H and critical values from this distribution can be used for a test with asymtotically correct size. Notice that the degrees of freedom equal the number of autocorrelations that are hyothesized equal to zero. 8. onsider the linear regression model y t = + z t + y t + u t, u t m:d:s:, t = ; ; :::; : De ne the roerty of conditional homoskedasticity and roose a test of the null hyothesis that u t are conditionally homoskedastic. 9. What is an econometric model?. Assume that economic theory redicts a relationshi of the form E(y t jw t ; y t ; w t ; :::; y ; w ) = w t ; where w t is not observable. Instead, observations of a variable z t are available such that z t = w t + " t ; where " t denotes a measurement error satisfying E(" t ) =, V ar(" t ) = " > and E((y t w t )" t ) = : Show why OLS can not be alied to estimate consistently by regressing y t on z t. What is the robability limit of the OLS estimator of under stationary ergodicity?. Let (y t ; x t ), t = ; :::;, be a random samle from a distribution such that y t = x t + bx t + " t with Ex t = E(" t jx t ) =, Ex 4 t < and E(" t jx t ) = (; ). Here y t ; x t and " t are scalars.
Let ^ be the OLS estimator obtained by regressing y t on x t. (a) What value does ^ estimate consistently? Answer. By general theory, ^ estimates consistently the value identi ed by the moment condition Efx t (y t x t )g = ; or, uon inserting y t = x t + bx t + " t, Efx t ((b )x t + " t + x t )g = : he general theory is alicable because the data (y t ; x t ) is iid and su ciently high moments exist for the LLN to hold. As E(x t " t ) = E x t E[" t jx t ] =, the identifying condition for becomes from where as long as Ex 4 t 6=, i.e., P (x t 6= ) >. Efx t ((b )x t + x t )g = () ( b) Ex 4 t = Ex t ; = b + Ex t Ex 4 t In the following, let Ex t =, Ex 8 t < and E" 4 t <. (b) Find the asymtotic distribution of ^ (i.e., the limit distribution of ^ aroriately centred and normalised such that a non-degenerate limit obtains). Answer. If Ex t = ; then ^ P! b by the conclusion of art (b). Moreover, we can write y t = bx t + e t, where e t = x t + " t is such that regressor*error = x t e t = x t + x t " t is iid with zero mean (E(x t + x t " t ) = Ex t + E x t E[" t jx t ] = ) and nite variance E(x 4 t e t ) = E(x t + x t " t ) = Ex 6 t + E(x 4 t " t ) + E(x 5 t " t ) (5) = Ex 6 t + E[x 4 t E(" t jx t )] + E[x 5 t E(" t jx t )] = Ex 6 t + Ex 4 t : herefore, the L alies and general theory yields the following limit distribution with variance in the sandwich form: w! ^ b N ; Ex6 t + Ex 4 t (Ex 4 t ) : (6). onsider a bivariate random samle (y t ; x t ), t = ; :::; ; where y t = x t +" t with E (" t jx t ) = ; E " t jx = (; ) and Ex 4 t <. P xt " (a) What is the limiting distribution of t t= x t " t xt " Answer. Under the random samle assumtion, t x t " t xt " E t x t " t E(xt " = E t jx t ) E(x t " t jx t ) as!? Justify. is an i.i.d. sequence with mean xt E(" = E t jx t ) x t E(" t jx t ) = by the law of iterated exectations, and variance matrix x E t " t x t " t x x t " t x 4 t " = E t E(" t jx t ) x t E(" t jx t ) t x t E(" t jx t ) x 4 t E(" t jx t ) = x E t x t x t x 4 t similarly. he variance matrix is nite since Ex 4 t <. herefore, from the L for iid sequences, X t= xt " t x t " t d! N ; x E t x t x t x 4 t as!. he same result could be obtained using the L for martingale di erence sequences. (b) If ^ is the OLS estimator obtained by regressing y t on x t and ^e t = y t ^xt are the regression P residuals, what is the limiting distribution of t= x t^e t as!? Justify. (Hint: It is not the same as the limiting distribution of P t= x t" t.)
Answer. It is a well-known fact that OLS residuals are orthogonal to the regressors, so P t= x t^e t is a sequence of zeroes and converges to zero. In robabilistic terms, the limit distribution is a oint mass at zero, i.e., the distribution of a random variable assuming the value of zero with robability one. he veri cation of the fact that residuals are orthogonal to x t is straightforward. As ^e t = y t ^xt and ^ = ( P t= x t ) ( P t= x ty t ), it holds that X t= x t^e t = = X t= X t= x t y t ^ X t= x t x t y t P t= x ty t P t= x t X t= x t = X t= x t y t X t= x t y t = :. Let ^ = (^ ; :::; ^ k ) be an estimator of a arameter R k, k. Let it be known that d! ^ N ; Ik + as!, where Ik is the k k identity matrix and R k is a xed non-random vector. onsider the hyothesis H that the entries ;i, i = ; :::; k, of = ( ; ; :::; ;k ) satisfy ; ; = ; ; = ::: = ;k ;k : In words, H is the hyothesis that the entries of form an arithmetic rogression. (a) For testing H, suggest a test statistic which is asymtotically -distributed when H is true. How many degress of freedom does the limit distribution under H have? Answer. he null hyothesis H is that the k equalities ;i ;i+ ;i+ =, i = ; :::; k, hold. It can be written in matrix form as R = where R = (r ij ) is the following uer triangular (k ) k matrix of full rank k : R = B @ ::: ::: :::.......... with r ii = (i = ; :::k), r i;i+ = (i = ; :::; k ), r i;i+ = (i = ; :::; k ) and all the remaining entries equal to zero. From the fact that d! ^ N ; Ik + and the ontinuous maing theorem [M] it follows that d! R^ R N ; R[Ik + ]R : () Again from d! ^ N ; Ik + P and the M it follows that ^! and, by alying the M once more, R[I k + ^^ ]R P! R[Ik + ]R. he matrix R[I k + ^^ ]R is invertible because det(i k + ^^ ) and R is full rank, so by recalling () and alying... the M it is concluded that fr[ik + ^^ ]R g = R^ R d! N (; Ik ) A and R^ R fr[ik + ^^ ]R g R^ R d! (k ); (8) where k is the row dimension of R (the number of scalar restrictions). Observe that the revious convergence holds indeendently of whether H holds. Under H it holds that R =, so the revious convergence secialises to W := ^ R fr[i k + ^^ ]R g R^ d! (k ): he statistic W can be suggested as ful lling the requirements of item (a), and as discussed next, also those of item (b). An even simler statistic obtains by substituting R by also in the limiting variance in (). Seci cally, under H, () simli es to R^ d! N (; RR ), so a test of H could be based on the statistic ^ R (RR ) R^ which under H has (k ) limiting distribution, similarly to W.
(b) Say somebody suggests instead a statistic of the form = ^ + ^ ^ q ; ^ + 4^ + ^ + ^ ^ 4^ ^ 4^ ^ + 6 which under H satis es the convergence d! jn (; ) j as!. Make sure that the statistic you suggest in (a) is referable to whenever k > and justify what the advantage of your statistic is. Answer. he motivation for is as follows. Let r = (; ; ; ; :::) be k with r i = ; i 4: hen, as in item (a), it holds that fr[ik + ^^ ]r g = r^ r d! N (; ) : Here r^ = ^ +^ ^, r = ; + ; ; and r[i k +^^ ]r = ^ +4^ +^ +^ ^ 4^ ^ 4^ ^ +6, so ^ + ^ ^ ( ; + ; ; ) d q! jn(; )j: (9) ^ + 4^ + ^ + ^ ^ 4^ ^ 4^ ^ + 6 Under H it holds that ; + ; ; = ; so one rediscovers that! d jn (; ) j under H, as you were told from the outset. For k = it holds that W =, so it makes no di erence whether one uses W or for a test of H. For k >, however, has the disadvantage that it gives rise to a test with no ower against certain alternatives to H. Seci cally, is only sensitive to violations of the restriction ; + ; ; =, though not of the remaining k restrictions. For examle, if ; + ; ; = but ;4 + ; ; 6=, then H is not true. Nevertheless, because of (4), it is still true that! d jn (; ) j. herefore, as the samle size increases, a test based on will reject the wrong H with robability aroaching the size of the test, which is usually chosen very small. Instead, it would be referrable that the test rejects with robability aroaching whenever H is violated. esting with W has this roerty. Seci cally, it holds that W R fr[i k + ^^ ]Rg = R R^ R fr[ik + ^^ ]Rg = R^ R : Here R fr[i k + ^^ ]Rg = P R! R fr[i k + ]Rg = R by the M, so R fr[i k + ^^ ]Rg = P R! whenever R 6=, whereas (8) holds indeendently of whether H is true or not. Hence, W! P whenever R 6= and, for any critical value c:v:, a test rejecting H for W > c:v: will in fact reject it with robability aroaching one whenever R 6= : Summarising, W gives rise to a consistent test whereas does not. 4. Let f(y t ; x t )g tn be a random samle from a joint distribution such that y t = + x t + x t "t ; where " t is indeendent of x t and additionally E" =, E" = (; ), Ex 4 t <. Moreover, ; ; and are unknown arameters. hese assumtions imly the reresentation for aroriately de ned e t. (a) Find E(e t jx t ) and E(e t jx t ): y t = + x t + x t + e t () hroughout, e t = + x t + x t ("t ) is i.i.d. as a transformation of the i.i.d. (y t ; x t ). We nd that E(e t jx t ) = E( + x t + x t ("t )jx t ) = + x t + x t E("t jx t ) = + x t + x t E("t ) =
because " t is indeendent of x t with E" t =, and E(e t jx t ) = E( + x t + x t ("t ) jx t ) = + x t + x t E(("t ) jx t ) = + x t + x t E[("t ) ] = + x t + x t () again by indeendence and because E[(" t ) ] = V ar" t = E" t (E" t ) : (b) Formulate the hyothesis of conditional homoskedasticity in () as a linear restriction on ( ; ; ) and roose a test statistic, with asymtotic distribution free of unknown arameters under the null, for testing it. Justify the asymtotic distribution. In view of (4), the hyothesis of conditional homskedasticity is = = (unless x t = const which is excluded in what follows), or equivalently, R = with R = and = ( ; ; ) : Let ^ be the OLS estimator based on (). moments (E" t < ; Ex 4 t < ) it holds that Under the random samle assumtion with enough ^ d! N ; [E(X X )] E(X X e )[E(X X )] ; () where X t = (; x t ; x t ). Under the null of homoskedasticity the limiting variance simli es, so ^ d! N ; [E(X X )] E(e ) : Still under the null we can relace R by and nd further that d R^! N ; R[E(X X)] R E(e ) ;! = X ^ 4R X t Xt R 5 R^! d N (; I ) t= ^ R 4 ^! R X X t Xt R 5 t= R^ d! () where P t= X txt is a consistent estimator of E(X X) and the residual variance ^ is a consistent estimator of E(e t ). he sought test statistic is on the l.h.s. of the dislay above. 5. onsider the AR() rocess (autoregressive of order one) y t = y t + " t ; t N; () where " t are i.i.d. ;, jj < and y is given a distribution such that y t is stationary. For yt estimation of based on a samle of size, consider the vector of instruments z t = y t and the moment condition E(z t " t ) = : (a) Is the rank condition for identi cation satis ed? Why? In class notation, x t = y t ; so for > we nd that zx = E(z t x t ) = Ey t E(y t y t ) = has rank, equal to the dimension of x t : his means that the rank condition is satis ed. (b) Show that the asymtotically otimal choice of a weighting matrix for GMM estimation W = S (in class notation) reduces here to W = 4 : 4
First, S = E(z t z t" t ) = E(z t z t)e(" t ) = E(z t z t) using the indeendence of z t and " t, and second, E(z t zt) Eyt = E(y t y t ) E(y t y t ) Eyt = so W = S = 4 = 4 (c) Use the asymtotically otimal choice of W from art (b) to evaluate whether GMM has an asymtotic advantage over OLS (it could be useful to nd an exlicit exression for the GMM estimator with this W or with a consistent estimator ^W ). How would you roceed to evaluate whether GMM has a nite-samle advantage? It is known that (^OLS )! d N ; : ; (make sure you can derive this) and (^GMM ) d! N ; ( xz W zx ) ; where xz W zx = ( ) = ; so also (^GMM ) d! N ; : hus, in terms of e ciency, there is no asymtotic di erence between OLS and GMM. In nite samles the relative erformance will deend on the choice of ^W. If ^W = ^ P P y P t P yt y t yt y t y ; t where ^ is some estimator of and P P y P t P yt y t yt y t y t P! Ey t E(y t y t ) E(y t y t ) Ey t then GMM and OLS roduce numerically equal estimators: S xz ^W Szx = ^ P P P P y t yt y y t yt y t t P P yt y t y t = ^ X y t ; S xz ^W Szy = ^ P y t = ^ X yt y t ; ; P y P t yt y t P P P yt y y t yt y P t t P P P yt y t yt y t y t yt y t ^ GMM = (S xz ^W Szx ) S xz ^W Szy = X y t X yt y t = ^ OLS : For other choices of ^W the relative nite-samle erformance of GMM and OLS can be studied by simulation. (d) Using the asymtotically otimal choice of W from art (b), which moment condition (or linear combination of moment conditions) is identifying and which one is overidentifying? What is the meaning of the restriction tested by the J test? Is this restriction satis ed for the AR() rocess under discussion? he identifying condition is xz W E(z t (y t y t )) = ; or equivalently, ( ) E(yt (y t y t )) E(y t (y t y t )) E(y t (y t y t )) E(y t (y t y t )) = ; = ; E(y t (y t y t )) = ; 5
which is the same as for OLS. he identi ed true value is the solution of the equation E(y t (y t E(ytyt ) y t )) =, i.e., = = or (y Eyt t ; y t ). It equals the autoregressive arameter of the AR() rocess. Otimal GMM does not use the overidentifying condition E(y t (y t y t )) = for estimation. esting this condition amounts to a test of E(y t (y t y t )) = ; or equivalently, E(y t " t ) =, which is a true hyothesis. Since y t y t is the residual rocess, this can be seen as a test of the hyothesis that there is no unaccounted second-order autocorrelation in the residual. If there were, y t wouldn t be AR(). his is seen in question (e). (e) If y t is generated by y t = y t + " t ; t N; (4) with y and y given the stationary distribution, but one insists on estimating () with the same instruments z t and the same W as before, what value of is identi ed as "true" by the identifying (linear combination of) moment condition(s) and what is the asymtotic behavior of the J statistic? We now have Eyt = and Ey t y t = ; so the identifying condition E(y t (y t y t )) = identi es as true value = E(ytyt ) Ey t = or (y t ; y t ) = : esting the overidentifying condition E(y t (y t y t )) = reduces to E(y t y t ) = ; which is equivalent to = in (4). If =, the J statistic is asymtotically (), whereas if 6=, it is unbounded in robability and the associated test rejects with robability aroaching one. Hence, the J test erforms as a test of goodness of the AR() seci cation. 6. Let (y t ; x t ) ; t = ; :::;, be a random samle from a distribution such that y t = x t + " t ; (5) where and x t are scalars, E(x t ) >, E (x t " t ) = and E(" t jx t ) = const. onsider GMM estimation of based on the moment conditions E(x t (y t x t )) = ; E(x t (y t x t )) = : - heck the rank condition for identi cation. - Find an asymtotically otimal weighting matrix W. - With resect to this W, determine the identifying and the overidentifying linear combination of moment conditions. - an the J-test can rovide evidence whether (5), aart from reresenting a linear rojection, is also a regression equation (i.e., E (y t jx t ) = x t )? Exlain. 6