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14.384: Time Series Analysis. Bank of sample problems for 14.384 Time series Disclaimer. The problems below do not constitute the full set of problems given as homework assignments for the course. Some of the problems are well-known folklore, some were inspired by the problem sets given at diﬀerent times at Harvard, Upenn and Duke. I am thankful to Jim Stock, Frank Schorfheide and Barbara Rossi for giving me access to their course materials. 1. Transforming AR(p) to MA. If a p -order autoregressive process φ(L)yt = εt is stationary, with moving average representation yt = ψ(L)εt , show that 0= p � φj ψk−j = φ(L)ψk , k = p, p + 1, . . . j=0 i.e., show that the moving average coeﬃcients satisfy the autoregressive diﬀer­ ence equation. 2. Sims’ formula for spectrum. Assume that we have a sample {yt , xt }Tt=1 from in­ � j ﬁnite distributed lag model yt = B(L)xt +et , B(L) = ∞ j=1 bj L with absolutely � summable coeﬃcients |bj | < ∞ (here et is a white-noise, xt is stationary and weakly exogenous). Assume that one estimates (misspeciﬁed) model with q lags, a1 , ..., Þ aq . As the sam­ that is, he regresses yt on to xt−1 , ..., xt−q−1 and obtains Þ ple size increases to inﬁnity (but q is kept constant), the estimated coeﬃcients p converge to some non-random limits: Þ aj → aj . Let A(L) = a1 L + ... + ap Lp . Show that A(·) is a solution to the following problem: � � � � 1 π � A(e−iω ) − B(e−iω ) SX (ω) A(eiω ) − B(eiω ) , min a1 ,...,aq 2π −π where SX (·) is the spectrum of the process xt . That is, one minimizes the quadratic form in the diﬀerences between true and estimated polynomial, as­ signing the greatest weights to the frequencies for which spectral density is the greatest. 1 3. Spectrum and ﬁlters. This is your ﬁrst empirical problem. Choose a software package you feel comfortable using (I would recommend MatLab).You may use any users-written codes you ﬁnd on Internet. Always make sure that the code is doing what you think it is doing. Please, do not forget to cite whatever you are using. (i) Download quarterly values of Real GDP for the US from Mark W. Watson personal web-site (you may use any other aggregate macro time series from any other source if you wish. Economic Database (FRED II) maintained by the Federal Reserve Bank of St. Louis is a fantastic source ). (ii) Deﬁne the growth rate for real GDP. Estimate and plot spectrum for the growth rate. Discuss the graph. Find which peak in the spectrum corre­ sponds to business cycles. (iii) Use the following three cycle removing devices: a) run the OLS to detrend the series ; b) use Prescott-Hodrick ﬁlter; c) apply Baxter-King ﬁlter. (iv) Re-estimate spectrum for all series after applying each of the three proce­ dures. Draw spectrum functions. Discuss the diﬀerences. Note. As in real life empirical research, you will need to make a lot of choices while performing the task, such as choosing lag length, kernel func­ tion, and so on. Try to be reasonable, always check whether you results are sensitive to these choices. Also check original papers for suggestions. 4. Factor model and Principle components. Let X = (Xit )i=1,...,N,t=1,...,T is T × N matrix of observations. The matrices Λ(N ×k) and F (T ×k) are both unknown. Factor model could be written as an N -dimension time series with T observa­ tions: Xt = ΛFt + et , where Xt = (X1t , X2t , ..., XN t )� , Λ = (λ1 , ..., λN )� and et = (e1t , e2t , ..., eN t )� . Alternatively, it can be written as a T -dimension system with N observations: Xi = F λi + ei , where Xi = (Xi1 , ..., XiT )� , F = (F1 , ..., FT )� , and ei = (ei1 , ..., eiT )� . The method of principle components minimizes 1 V (k) = min Λ,F N T N � T � (Xit − λ�i Ft )2 i=1 t=1 (a) Write down the ﬁrst order condition for minimization over Λ. Concentrate out Λ. 2 (b) Assume the normalization F � F/T = Ik . Show that minimization problem is equivalent to maximizing trace of F � (XX � )F . (c) Argue that F consists of the linear subspace containing k eigenvector cor­ responding to the k largest eigenvalues. (d) What are the estimates of factor loadings and common component? 5. Subsampling with of nearly unit root process. Assume that you have a sample {x1 , ..., xT } of size T from an AR(1) process with the autoregressive coeﬃcient 0 < ρ ≤ 1. The goal is to construct an asymptotic conﬁdence set for ρ. Subsampling ( Romano and Wolf, Econometrica 2001) is the following procedure. Step 1. Regress xt on its lag and calculate the OLS estimate of ρÞ and variance σ Þρ2Þ. Step 2 Choose a subsample size bT < T and let b be an index changing be­ tween 1 and T − bT . For each value of b consider a subsample Zb = {xb , xb+1 , ..., xb+bT } of the size bT from the initial sample. For each block Zb run OLS regression to get the t-statistics, tb = ρÞb −ρÞ . ÞρÞb σ −bT Step 3 Order statistics {tb }Tb=1 in ascending order and get α/2 and 1 − α/2 quantiles (q1 and q2 ) of this distribution. The conﬁdence set for ρ is C(x) = [ρÞ − q2 σ ÞρÞ, ρÞ − q1 σ ÞρÞ]. The purpose of this problem is to understand the asymptotic coverage of the described procedure. Let t(T, ρ0 ) = ρÞ−ρ0 σ ÞρÞ be the t-statistics for testing H0 : ρ = ρ0 with the full sample. The described procedure uses an approximation of unknown distribution of t(T, ρ0 ) by the distribution of t-statistic in subsample tb (bT ) = ρÞb −ρÞ , σ ÞρÞb here ρÞb is calculated for a subsample of size bT , and ρÞ - for the whole sample. The distribution of tb (bT ) could be simulated (it is done in Step 3). Assume known that simulated in Step 3 distribution (quantiles) of tb (bT ) are uniformly close(converge) to the theoretical distribution(quantiles) of tb (bT ). Assume that bT → ∞ and bT /T → 0 as T → ∞. (a) Let the true value 0 < ρ0 < 1 be ﬁxed while T → ∞. What is the 3 limiting distribution of t(T, ρ0 )? What is the limiting distribution of tb (bT )? Calculate the limiting coverage limT →∞ Pρ0 {ρ0 ∈ C(x)}. (b) Now assume that we have a unit root, that is, ρ0 = 1. What is the limiting distribution of t(T, ρ0 = 1)? What is the limiting distribution of tb (bT )? Calculate the limiting coverage limT →∞ Pρ0 =1 {1 ∈ C(x)}. For the next steps use the following statement. Assume that ρT = 1 + cT /T . �1 w(t)dw(t) • If cT → 0 as T → ∞, then t(T, ρT ) ⇒ √0 � 1 2 . 0 w (t)dt • If cT → −∞ as T → ∞, then t(T, ρT ) ⇒ N (0, 1). (c) Let ρT = 1 + c/T, c < 0. What is the limiting distribution of t(T, ρT )? What is the limiting distribution of tb (bT )? What can you say about the limiting coverage limT →∞ PρT {ρT ∈ C(x)}? What is the intuition of your result? (d) Let ρT = 1 + c/bT , c < 0. What is the limiting distribution of t(T, ρT )? What is the limiting distribution of tb (bT )? What can you say about the limiting coverage limT →∞ PρT {ρT ∈ C(x)}? What is the intuition of your result? (e) Is the subsampling interval uniformly asymptotically correct? Explain. 6. Empirical exercise. PPP puzzle. Purchasing power parity (PPP) is “an em­ pirical proposition that, once converted to a common currency, national price levels should be equal” (Rogoﬀ, 1996). Even though almost no one believes in absolute PPP, most think that the real exchange rate tend toward PPP in the very long run. Main puzzle, however, is that the speed of convergence (mea­ sured as a half-life of a shock and estimated to be between three to ﬁve years) is too slow to be explained by nominal rigidities. This exercise is aimed to answer two questions: 1) is there any long-run conver­ gence of PPP ( rephrase: does real exchange rate possess a unit root); 2) what is the half-life of real exchange rate? 4 Take any currency and calculate a time series for log of real exchange rate (exchange rates and CPI for various countries provided on the course webpage). I call it xt . (a) Test whether xt has a unit root. Use augmented Dickey-Fuller (with lag length chosen according BIC) and Phillips-Perron test. Do this in two versions: including constant and including a linear trend. Do the data show evidence of a unit root? (b) Use Stock (Journal of Monetary Economics, 1991) method to construct a conﬁdence interval for the local to unity parameter. Assume that xt is AR(1) with a constant and use local to unity approximation for t-statistic. Tables are reported in Stock(1991). (c) Assume that xt = c + ρxt−1 + et . The half-life is deﬁned as a time needed for half of the shock to die. That is half-life = − log 2 . log ρ Transform the conﬁdence set you received in (b) to a conﬁdence interval for the half-life of deviations from PPP. How persistent are the shocks? 7. Simulated GMM. Suppose we wish to estimate an MA(2) process yt = µ + et + θ1 et−1 + θ2 et−2 where the et are iid N (0, σ 2 ) random variables. Although estimation is possible using ML, explain how you could estimate the parameters of the model using indirect inference. Also indicate how the models speciﬁcation can be tested using indirect inference. 8. Kalman ﬁlter of a long-run trend. Consider a model of a constant long-run trend αt = αt−1 , yt = αt + vt , vt ∼ i.i.d.N (0, σ 2 ). (a) Write down Kalman ﬁlter for the model with starting values α1|0 = a and P0|1 = p0 , then Pt|t = 5 p0 . 1 + tp0 /σ 2 Show also that the contribution of each additional observation to αT |T is negligible as T increases. (b) Show that as time horizon increases Kalman ﬁlter converges to a value independent of a and p0 . What is this value? (c) What is the value of αT |T if we have uninformative prior p0 → ∞? Readings: Christiano, Eichenbaum and Vigfusson (2004) “What happens after a technology shock?” unpublished manuscript. Gali “ Technology, Employment and the Business Cycle: Do technology shocks explain aggregate ﬂuctuations?”, AER 1999. Romano J.P. and Wolf M.(2001): ”Subsampling Intervals in Autoregressive Mod­ els with Linear Time Trend,” Econometrica, 69(5), 1283-1314. Stock, J. (1991). ”Conﬁdence intervals for the largest autoregressive root in US macroeconomic time series,” Journal of Monetary Economics 28, 435-459. 6 MIT OpenCourseWare http://ocw.mit.edu 14.384 Time Series Analysis Fall 2013 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.