Economics Department LSE. Econometrics: Timeseries EXERCISE 1: SERIAL CORRELATION (ANALYTICAL)

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1 Economics Department LSE EC402 Lent 2015 Danny Quah TW A x7535 : Timeseries EXERCISE 1: SERIAL CORRELATION (ANALYTICAL) 1. Suppose ɛ is w.n. (0, σ 2 ), ρ < 1, and W t = ρw t 1 + ɛ t, for t = 1, 2,.... Suppose EW 0 = 0, EW 2 0 = (1 ρ2 ) 1 σ 2, and W 0 is uncorrelated with all ɛ t, t = 1, 2, 3,.... Prove that for all t = 1, 2, 3,..., we have EW t = 0 and Var W t = (1 ρ 2 ) 1 σ 2. Is W covariance stationary? Is W stationary? 2. Suppose ɛ is w.n. (0, σ 2 ) and W t = γ 0 ɛ t + γ 1 ɛ t 1 + γ 2 ɛ t 2. Prove that the covariogram of W satisfies (γ γ2 1 + γ2 2 ) σ2 for s = 0; (γ 0 γ 1 + γ 1 γ 2 ) σ 2 for s = ±1; G W (s) = γ 0 γ 2 σ 2 for s = ±2; 0 otherwise. Is W covariance stationary? Is W stationary? 3. Let X and Y be zero-mean covariance stationary processes with covariograms G X and G Y respectively. Suppose that X and Y are uncorrelated at all leads and lags. Show Z t = X t + Y t = G Z (s) = G X (s) + G Y (s) for all s = 0, ±1, ±2,..., i.e., covariograms of orthogonal processes add.

2 4. Suppose ɛ and ν are covariance stationary processes that are orthogonal, i.e., Eɛ t ν s = 0 for all t, s. Define X to be the sum of ɛ and ν, i.e., X t = ɛ t + ν t. (a) Denoting covariograms by G, prove that G X = G ɛ + G ν, i.e., covariograms of orthogonal processes add. (b) (If convenient, use the previous result in answering this.) Denoting spectral densities by S prove that S X = S ɛ + S ν, i.e., spectral densities of orthogonal processes add. 5. Suppose ɛ and ν are unit variance w.n. processes uncorrelated with one another. We observe W t = ɛ t ɛ t 1 + φν t (a) Find the covariogram of W as a function of φ. (b) A colleague points out that if φ = 0, the remaining component of W, that in ɛ s, is just an MA(1). For φ 0, he says, this is therefore an MA(1) contaminated by error. The result, therefore, cannot be an MA(1) process. Do you agree? Why? How does your answer vary with the value of φ? (c) The same colleague says the error contamination when φ 0 is unfortunate, as uncontaminated data are always better than contaminated data; moreover, since ν is uncorrelated with ɛ, the variance of W is larger with φ 0. Calculate the spectral density of W as a function of φ. Do you agree with your colleague? Why? (Hint: Central Limit Theorem.) 6. A researcher estimates for a covariance stationary Y the model Y t = 0.5Y t 1 + ɛ t 2.01ɛ t 1, ɛ w.n. (0, σ 2 ) with the standard errors on the estimated coefficients very small, and enlarging the model with additional lags fails to improve the fit. This 2

3 researcher s colleague directly estimates the spectral density of Y and finds that spectral density approximately flat: She concludes Y is approximately w.n.. The first researcher, however, suggest that the spectral density estimate must be incorrect since his own estimates suggest Y is not w.n. but instead ARMA(1,1). Show that, in fact, both are correct. (Hint: Obviously, 0.5 1/2.01.) 7. Suppose X t = ɛ t + θ 1 ɛ t 1 + θ 2 ɛ t 2 where θ 1 = ζ 1 + ζ 2 θ 2 = ζ 1 ζ 2 ɛ w.n. (0, 1). (a) Derive the covariogram of X, i.e., G X (s), for s = 0, ±1, ±2,... in terms of (θ 1, θ 2 ). (b) Suppose ζ 1 = 1 and ζ 2 < 1. Establish that X has covariogram sum 0, i.e., G X(s) = 0. (c) Consider X t = X t + η t where η is w.n. (0, σ 2 ), with σ 2 > 0, and uncorrelated with ɛ at all leads and lags. Argue that now regardless of the value of ζ 1 the process X satisfies a central limit theorem, i.e., for some ν 2 > 0. T 1 2 T X t t=1 L N(0, ν 2 ) 8. Consider the covariance stationary AR(1) process X t = ρx t 1 + ɛ t, ρ < 1, ɛ w.n. (0, 1). (a) Calculate the covariogram G X (s), s = 0, ±1, ±2,... in terms of ρ. (b) Given observations on X t, all integer t, a research assistant accidentally runs the regression X t = βx t+1 + ν t (i.e., instead of regressing X t on the lag X t 1, he regresses X t on future X t+1 ). Prove that OLS consistently estimates β = ρ. 3

4 (c) What is the relation between ν and ɛ? Show that ν is serially uncorrelated. 9. Consider the linear regression model Y t = X t β 0 + u t, with EX t u t = 0 (9.1) where OLS will be applied to estimate β 0. Suppose Y has mean zero and is covariance stationary. (a) Show that the covariogram G Y, with G Y (s) = E[Y t Y t s ], is symmetric, i.e., G Y (s) = G Y ( s). (b) For X t = Y t 1 the regression equation (9.1) represents an attempt to estimate a first-order autoregression regardless of the true underlying model for Y. In terms of G, what does the OLS estimator here for β 0 converge to? (c) Suppose instead X t = Y t+1, i.e., the regressor is future Y rather than past. Show that in this case regression equation (9.1) has the OLS estimator converge to the same value as when X t = Y t 1. (d) Suppose Y is truly a first-order autoregression, i.e., 10. Suppose Y t = β 0 Y t 1 + ɛ t, with ɛ w.n. (0, 1) and β 0 < 1. Show that ν t def = Y t β 0 Y t+1 is also serially uncorrelated. Be careful to note that ν t = (β 1 0 Y t Y t+1 ) β 0 ɛ t+1 β 0. X t = ɛ t + θɛ t 1 where ɛ w.n. (0, 1) (a) Derive the covariogram of X, i.e., G X (s), for s = 0, ±1, ±2,... in terms of θ. (b) Suppose θ = 1. Establish that X has covariogram sum 0, i.e., G X(s) = 0. 4

5 (c) Consider X t = X t + η t where η is w.n. (0, σ 2 ), with σ 2 > 0, and uncorrelated with ɛ at all leads and lags. Under what circumstances will s G X(s) = 0, i.e., the covariogram sum of X equal 0? 11. Prove that if the zero-mean covariance stationary X has moving average representation then and therefore X t = C j ɛ t j, ɛ w.n. (0, 1), j=0 s= ( ) 2 G X (s) = C j, s= j=0 G X (s) 0 with equality precisely when j=0 C j = Suppose X t = ɛ t + θɛ t 1, with ɛ iid N(0, 1). (a) Prove that X has covariogram G X (s) = E[X t X t s ] satisfying s= G X (s) = 1 + θ 2 + 2θ. (b) Suppose θ = 1. Prove that s= G X(s) = 0 and that X does not satisfy a Central Limit Theorem, i.e., T 1/2 T t=1 X t fails to converge to a nondegenerate normally distributed random variable as T. What is the distribution of T 1/2 T t=1 X t for fixed T? (c) Suppose θ = 0 and W = { 1 with probability 1/2; +1 with probability 1/2 5

6 independent of ɛ. Consider the timeseries Prove that EY t = 0; Y t = W + X t. G Y (s) = Cov(Y t, Y t s ) = { 2 for s = 0; 1 otherwise. Does Y satisfy a Law of Large Numbers? Sketch why or why not. 13. Suppose u t = ɛ t + θɛ t 1, ɛ w.n. (0, 1). (a) Derive the covariogram of u in terms of θ. (b) Suppose θ = 1. Establish that u has covariogram sum equal to 0. (c) Consider the regression model: y t = X t β + u t, where u t is as above, with θ = 1, and where X t = 1, i.e., the regressors are just a constant. What are the properties of the OLS estimator for β? 14. Suppose W = { W t : t = 0, 1,... } follows W t = ɛ t + θɛ t 1, ɛ w.n. (0, ν 2 ), θ > 1. (a) Find W s covariogram G W. (b) Suppose Y = { Y t : t = 0, 1,... } follows Y t = η t + φη t 1, η w.n. (0, σ 2 ), φ = θ 1 = φ < 1. Under what (if any) restrictions on σ 2 will the covariogram of Y equal that of W? 6

7 (c) Suppose that in W the w.n. ɛ is iid N(0, ν 2 ). Show that if then λ 1 T 1/2 λ 2 def = T W t t=1 s= s= G W (s) L N(0, 1) for T. 15. A researcher estimates for a covariance stationary Y the model Y t = 0.5Y t 1 + ɛ t 2.01ɛ t 1, ɛ w.n. (0, σ 2 ) with the standard errors on the estimated coefficients very small, and enlarging the model with additional lags fails to improve the fit. This researcher s colleague directly estimates the spectral density of Y and finds that spectral density approximately flat: She concludes Y is approximately w.n.. The first researcher, however, suggest that the spectral density estimate must be incorrect since his own estimates suggest Y is not w.n. but instead ARMA(1,1). Show that, in fact, both are correct. (Hint: Obviously, 0.5 1/2.01.) 16. A researcher decides he has seen enough comparison across the longrun growth of China and the US. Instead, he seeks to contrast their business cycle behaviour, and does so by focusing on the cyclical properties of the first difference of GDP. Denote this first difference variable X (subscripted US or China as needed.) For his data sample he has concluded that the spectral density of the US is single-peaked and tails off towards zero at high frequencies. In other words, the US shows serial correlation, more pronounced at certain intermediate cyclical frequencies but with little variation at high frequencies. Next he estimates an ARMA(2,1) model for China to be: X t = 0.91X t X t 2 + ɛ t 1.053ɛ t 1, ɛ w.n. (0, σ 2 ), with very small standard errors on the estimated coefficients, and where enlarging the model with more lags fails to improve the fit. (a) Assume for the moment that the coefficient on ɛ t 1 is zero: Describe in words the dynamic response of X to the disturbance ɛ. 7

8 (b) A rival researcher reports that for China the spectral density for X is basically flat, i.e., China s economy shows no serial correlation, or, in other words, no business cycles. Describe in words why the technical finding, that the spectral density is flat, implies this conclusion. (c) In light of the estimated equation above, can the rival researcher s finding be correct? Explain why. (If you wish, you can notice these numerical facts z 0.038z 2 = (1 0.95z)( z) with denoting approximately.) 8