Econometrics II - EXAM Answer each question in separate sheets in three hours. Let u and u be jointly Gaussian and independent of z in all the equations. a Investigate the identification of the following system of simultaneous equations y + γ y + δ + δ z = u y + δ + δ z = u. How does your answer change if δ = 0 but this is ignored? If some of the equations are not identified, propose a minimum set of additional restrictions that would exactly identify them. b Obtain the reduced form for the following system with nonlinear endogenous variables, y + γ y + δ + δ z = u log y + δ + δ z = u, and check that it produces the following conditional expectations { } E y z] = γ exp σ δ δ z δ δ z { } E y z] = exp σ δ δ z where σ = V ar u ] by using that for a Gaussian r.v. X N 0, σ we have that E e X] = exp { σ}. c Analyze the identification of the two equations -, proposing appropriate instrumental variables IV and studying the corresponding moment conditions needed. d If δ where known, which one would be your optimal choice of IV s for the first equation? Provide the asymptotic variance of the SLS estimates for this equation. e Analyze the identification when δ = 0.
. Consider the time series model wherein y t = βy t + v t v t = θe t + e t is a sequence of disturbances generated by a first-order moving average process which is driven by a white-noise sequence e t of independently and identically distributed random variables with zero mean and variance σ e. a Assuming that β < obtain the first three elements of the autocovariance sequence of y t, γ y 0, γ y, γ y. For which values of θ is y t stationary? b Show that the estimate of β obtained by applying the ordinary least-squares procedure on the first equation, would tend to the value of β + θ β / + θ β + θ as the size of the sample increases. c Propose a test of H 0 : θ = 0. d Imagine that the model y t = β y t + β y t + error t is fitted to the data by OLS. What values would you expect to obtain for β and β in the limit, as the size of the sample increases indefinitely, when β = 0? e If θ 0 were known, propose a consistent estimate of β and find its asymptotic distribution.
3. Consider the following nonlinear regression model, y = h z; θ 0 + v with E y z] = h z; θ 0 and usual identification conditions hold. a Consider the estimation by OLS of a misspecified linear regression model y = z β + u and find the solution for β that minimizes the variance of u. Would the OLS estimate of β be consistent for it? And for θ 0? b Consider the following linearization around an arbitrary value θ, and set the following linear regression model, where y = h z; θ 0 + v 3 h z; θ + h z; θ θ θ 0 θ + v y = z θ 0 + error, 4 y = y h z; θ + h z; θ θ θ, z = h z; θ. θ Assuming that the approximation in 3 is perfect if we include a further term a = θ 0 θ h z; θ θ θ θ 0 θ, study the probability limit of the OLS estimate θ n of θ 0 in 4. c Show that if θ p θ 0, then θ n p θ 0 under standard identification assumptions, and that θn corresponds to a Gauss-Newton iteration of Nonlinear Least Squares NLS starting with θ = θ. d Assume that h z; θ = h z θ and that z θ = z θ + z θ. Consider testing the hypothesis H 0 : θ 0 = 0, under the condition that E v z ] = σ, by means of a Wald test based on NLS estimates. Provide a test statistic and the asymptotic distribution under the null. e Find the distribution of the restricted estimate of θ 0 under H 0 and compare it with that of the unrestricted one. Propose now a Lagrange Multiplier test for H 0 and show that this amounts to a linear regression of restricted NLS residuals on an appropriate set of regressors. 3
Econometrics II - EXAM SOLUTIONS. Let u and u be jointly Gaussian and independent of z in all the equations. a Investigate the identification of the following system of simultaneous equations y + γ y + δ + δ z = u y + δ + δ z = u. nd equation: is a reduced form, always identified. st equation: the order condition is not satisfied, no omitted exogenous variables, is not identified in any case. How does your answer change if δ = 0 but this is ignored? If this is ignored and not used, it would not help. If it is known, then eq. will be identified if δ 0. If some of the equations are not identified, propose a minimum set of additional restrictions that would exactly identify them. Alternative conditions would be that it is known that δ = 0, or that δ = 0, or that Cov u, u = 0. b Obtain the reduced form for the following system with nonlinear endogenous variables, y + γ y + δ + δ z = u log y + δ + δ z = u, and check that it produces the following conditional expectations { } E y z] = γ exp σ δ δ z δ δ z { } E y z] = exp σ δ δ z where σ = V ar u ] by using that for a Gaussian r.v. X N 0, σ we have that E e X] = exp { σ}. Using that y = exp {u δ δ z} the reduced form is given by y y ] = u γ exp {u δ δ z} δ δ z exp {u δ δ z} ], so that { } E y z] = exp { δ δ z} E exp {u } z] = exp σ δ δ z. c Analyze the identification of the two equations -, proposing appropriate instrumental variables IV and studying the corresponding moment conditions needed. The second equation is still identified, just defining y = log y. 4
To identify the second equation we can now use E y z] and expand exp { δ z} in power series, so any higher moment of z could be an IV for y in eq.. Set Z =, z, z and X =, z, y. Then we need rank {E ZX ]} = 3, where, a = exp { σ } δ, z y E ZX ] = E z z zy = E z z 3 z y z a exp { δ z} = E z z za exp { δ z} z z 3 z a exp { δ z} z exp { σ δ δ z } z z z exp { σ δ δ z } z z 3 z exp { σ δ δ z } d If δ where known, which one would be your optimal choice of IV s for the first equation? Provide the asymptotic variance of the SLS estimates for this equation. In general the optimal IV s are Z = Ω z E X z] =E X z] =, z, E y z]. Therefore in this case the optimal choice would be to take as extra IV exp { δ z} itself, so Z =, z, exp { δ z} and z y E Z X ] = E z z zy exp { δ z} z exp { δ z} exp { δ z} y z exp { σ δ δ z } = E z z z exp { σ δ δ z } exp { δ z} z exp { δ z} exp { δ z} exp { σ δ δ z } z a exp { δ z} = E z z az exp { δ z}, exp { δ z} z exp { δ z} a exp { δ z} exp { δ z} which produces almost the same result as OLS. e Analyze the identification when δ = 0. In this case E ZX ] = E. z z z za z z 3 z a and clearly the first and third columns are proportional, so rank is at most. Identification relies on z appearing in the second equation.. Consider the time series model wherein y t = βy t + v t v t = θe t + e t is a sequence of disturbances generated by a first-order moving average process which is driven by a white-noise sequence e t of independently and identically distributed random variables with zero mean and variance σ e., 5
a Assuming that β < obtain the first three elements of the autocovariance sequence of y t, γ y 0, γ y, γ y. For which values of θ is y t stationary? We can write, since β <, y t = β j v t j, j=0 so that if v t is covariance/strict stationary, so is y t. But v t is a MA on iid w.n., so it is always s.s. and cov. stationary. Then γ y 0 = V ar y t ] = β V ar y t ] + V ar v t ] + βcov y t, v t ] = { σ β e + θ + βθσ } e = σ { e + θ β + βθ } since Cov y t, v t ] = Cov β j v t j, v t = Cov β 0 ] v t 0, v t = Cov vt, v t ] = θσ e. Next, and j=0 γ y = Cov y t, y t ] = βv ar y t ] + Cov y t, v t ] = βγ y 0 + Cov y t, v t ] = β σ { e + θ β + βθ } + θσ e { β = σ e + θ β + βθ ] } + θ γ y = Cov y t, y t ] = βcov y t, y t ] + Cov y t, v t ] = βγ y + 0 = βγ y. b Show that the estimate of β obtained by applying the ordinary least-squares procedure on the first equation, would tend to the value of β + θ β / + θ β + θ as the size of the sample increases. We find that or p lim ˆβ = β + Cov y t, v t V ar y t = β + θσ e γ y 0 = β + p lim ˆβ = Cov y t, y t V ar y t = βγ 0 + θσ e γ 0 = γ γ 0 θ β + θ θ + β = β + θ β + θ θ + β. 6
c Propose a test of H 0 : θ = 0. If θ = 0, then the OLSE of β is consistent, so that the residuals ˆv t = y t ˆβy t, should be approximately uncorrelated. This can be tested using uncorrelation tests based on ˆρˆv j, such as Box-Pierce. Alternatively, LM tests could be derived for an MA model, which implies testing for ρ = 0. d Imagine that the model y t = β y t + β y t + error t is fitted to the data by OLS. What values would you expect to obtain for β and β in the limit, as the size of the sample increases indefinitely, when β = 0? We have that in this case y t is just a MA, γ y 0 = σ { e + θ } γ y = σ eθ γ y = 0, so that the OLSE will converge to the corresponding population parameters, β γ = y 0 γ y γ y β γ y γ y 0 γ y + θ θ θ = θ + θ 0 = + θ + θ θ θ θ θ + θ 0 + θ = + θ θ θ θ, and in particular β β = 0. e If θ 0 were known, propose a consistent estimate of β and find its asymptotic distribution. The idea would be to implement a GLS estimate, so that y t = βy t + v t and the vt, t =,..., T are uncorrelated. Note that in this case v t MA AR with coefficients α j = θ j as far as θ <, so vt depends on v j, j =,..., t, and when t, vt e t. Then this regression is asymptotically equivalent to regress where y t = j=0 θj y t j. 3. Consider the following nonlinear regression model, y t = βy t + e t y = h z; θ 0 + v with E y z] = h z; θ 0 and usual identification conditions hold. a Consider the estimation by OLS of a misspecified linear regression model y = z β + u 7
and find the solution for β that minimizes the variance of u. Would the OLS estimate of β be consistent for it? And for θ 0? As usual β 0 = arg min E y z β ] = E zz ] E zy] b = E zz ] E zh z; θ 0 ], which can be estimated consistently by OLS, while θ 0 = arg min E θ Of course β 0 θ 0 in general unless h z; θ = z θ. y h z; θ ]. b Consider the following linearization around an arbitrary value θ, y = h z; θ 0 + v 5 h z; θ + h z; θ θ θ 0 θ + v and set the following linear regression model, where y = z θ 0 + error, 6 y = y h z; θ + h z; θ θ θ, z = h z; θ. θ Assuming that the approximation in 5 is perfect if we include a further term a = θ 0 θ hz;θ θ θ θ 0 θ, study the probability limit of the OLS estimate θ n of θ 0 in 6. We have that θn = E n z z ] E n z y ] = E n z z ] E n z y h z; θ + h z; ] θ θ θ = E n z z ] h z; θ ] E n z θ θ 0 + a + v = E n z z ] E n z z ] θ 0 + E n z z ] E n z a] + E n z z ] E n z v] = θ 0 + E z z ] E z a] + E z z ] E z v] + o p = θ 0 + b + o p, where b = E z z ] E z a] is not necessarily negligible since θ could be far away from θ 0. c Show that if θ p θ 0, then θ n p θ 0 under standard identification assumptions, and that θn corresponds to a Gauss-Newton iteration of Nonlinear Least Squares NLS starting with θ = θ. In this case we can show that E n z a] = O p θ θ 0 E n z hz;θ θ θ ] = o p O p = o p under standard moment conditions, so θ n = θ 0 +o p. Then for Q n θ = E n θ Q n θ = E n y h z; θ h z; ] θ θ = E n y h z; ] θ θ θ z = E n y z ] + E n z θ z ] = E n z y ] + E n z z ] θ y h z; θ 0 ], 8
while θ θ Q n θ = E n h z; θ θ E n z z ] := G n h z; θ ] θ E n y h z; θ h z; θ ] θ θ so that the Gauss Newton iterations are θn = θ G n θ Q n θ = θ E n z z ] { E n z y ] + E n z z ] θ } = E n z z ] E n z y ]. d Assume that h z; θ = h z θ and that z θ = z θ + z θ. Consider testing the hypothesis H 0 : θ 0 = 0, under the condition that E v z ] = σ, by means of a Wald test based on NLS estimates. Provide a test statistic and the asymptotic distribution under the null. Wald test. We assume that the unrestricted estimate satisfies n / ˆθn θ 0 d N 0, E DE 0, σ E where E = p lim θ θ Q n θ 0 = E z z ] D = V ar θ Q n θ 0 = E v z z ] = σ E z z ] = σ E. Note that h z; θ / θ = h z θ / θ = ḣ z θ z, where z = ḣ z θ z. Then the test statistics is ḣ x = h x / x, so that W n = ṋ σ ˆθ n {Ê } where Ê :=E n z z ] and ˆσ is the residual variance., ˆθn d χ #θ e Find the distribution of the restricted estimate of θ 0 under H 0 and compare it with that of the unrestricted one. Propose now a Lagrange Multiplier test for H 0 and show that this amounts to a linear regression of restricted NLS residuals on an appropriate set of regressors. LM Test. In this case we need to study the restricted estimates of θ 0, so that where θ n = θn = arg min Q n θ H 0 0, θ n and we set Ln θ = Q n θ λ θ. Then so that Q n θn θ Q n θn θ n / θn θ 0 = λ = θ Q n θ 0 + = 0 = θ Q n θ 0 + d θ θ Q n θ 0 θn θ 0 + op θ θ Q n θ 0 θn θ 0 + op = θ θ Q n θ 0 n / N 0, E D E = N 0, σ E Q n θ 0 + o p θ, 9
while n / ˆθn θ 0 N Also 0, σ { E } and { E },, E unless E = E = 0. n / λ = n / θ Q n θ 0 + E θn θ 0 + op d = n / Q n θ 0 E E θ n/ Q n θ 0 + o p θ = I E E n / θ Q n θ 0 + o p N 0, σ I E E = N 0, σ { E E E = N 0, σ { E } E I E E E } and the test statistic is { LM n = n λ Ê } λ d χ #θ. Now note that λ = θ Q n θn = En y h z θ ] n h z θn = E n y h z θn ḣ z θn z ] = E n v θn ḣ z θn z ] which can be obtained -up to scale- from running a linear regression of the residuals v θn on the additional linearized regressors ḣ z θ n z. θ 0