Res Ec 797A Lecture 22 Fall 2011

Res Ec 797A Lecture 22 Fall 2011 - Res Ec 797A Forecasting...

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Unformatted text preview: Res Ec 797A: Forecasting Order Of Topics And Readings For Lecture 22 (November 21, 2011) Source Pagegs) Top'c Material Distributed Last Class But Not Covered Elder and Testing for Unit Roots: What ShOuld Students Be Taught? Kennedy paper (Distributed last class). Enders 213 Figure 4.13: A Procedure To Test For Unit Roots (2nd Edition) (Distributed last class). Enders 257 Figure 4.7: A Procedure To Test For Unit Roots (1St Edition) (This is the same as in the 2"" edition). (Distributed last class). BM 5 Outline of the paper by Elder and Kennedy: Strategies for testing for a unit root We will not cover Elder and Kennedy’s Cases 2 and 3. They are similar to Case I. BM handwritten Unaugmented restiicted model vs. unaugmented unrestricted model and unnumbered BM 1 SAS program: Applying Elder and Kennedy’s Testing Strategy for Case 1 BM 1 F-test results for my unaugmented model and for my augmented model GHJ 698-700 Textbook example of conducting the F-test (comparable to the BM results above) (Distributed last class) The next topic is VectorAutoregression (VAR). Before next class, please read the following pages in Enders and look at the following BM notes. Enders 264-272 Introduction to Vector Autoregression (VAR) BM 92 Transition from Enders’ coverage of VAR on pages 264-266 and 269-270. I have a complete example using two variables: . Y: Gross Domestic Product (GDP) . X: Investment BM 93-100 Mechanics‘of VAR systems using Y and X above Systems of Simultaneous Equations, Reduced Forms, and Vector Autoregression (VAR) - Define the two variables X{ and Y[ as follows: . YI: Gross Domestic Product (GDP) - X.: Investment ' Let these two variables be related to each other as follows: ' Y = (1] + OLZXt + a3Yu—l + U‘EXl—I + “I l . X = BI + BZYI + B3Yl-l + [34Xl-I + U! l - Econometricians call this 5% of two equations the structural fomi. - There is a problem(s) with estimating this system in its cun‘ent form : - X has a contemporaneous effect on Yt - Y1 has a contemporaneous effect on X‘ - Both Yt and X are endogenous. Other issues related to this are: . Their values are simultaneously determined. . They are functions of random disturbances. . Ordinary Least Squares (OLS) estimates are biased and inconsistent. . When the econometrician desires estimates of the stiuctural parameters, several things must be done before estimation, one of which is to obtain the reduced form. - Reduced fonn: Substitute Yt into XI, and then X into Y‘. The eventual result is: . Y1 =yl +y2Yt—I +YJXl—I + 8| . X1 = 51 + SZYl-l +Y3XH + (DI ' Notice how the reduced form eliminates any contemporaneous relationship between the left- hand side and the right-hand side. . The time—series econometrician calls this a Vector Autoregression system; VAR for short. - The time-series econometrician finds this form very useful for forecasting. 92 Mechanics of Vector Autoregression Recall the autoregressive model for Y whose value Yl depends on past values up to lag length P. - Assume the stochastic process to be stationary. (If it weren’t, some correction is done). ~ This is an AR(P) process. ‘ Form: Y : (plYt-l + ‘PzYt—z + + (pth—p + 8t 1 Recall: a, : white noise, i.e., E(e,) = 0 E(etes) = 0 for s¢t -' zero autocovariance E(stsl = 0: —~ constant variance Q: Suppose we know the order P (we have yet to find it). How can we get estimates of (pl, (p2, ..., (pp? A: Rather than use the nonlinear techniques previously suggested for time series models, why not use ordinary least squares? Notice that this equation is in the form of the linear statistical model. + + (9le + 8p“ Y = (plYp+l + + (ppY2 + 8W2 Y = ‘Pth—i + + (pthP + a, Note: - Each equation has “P” (p’s to be estimated. (In an actual problem, we have yet to determine P). - The subscript of the dependent variable matches its error. - The number of right-hand-side Y’s is P. ' The (1,1) element of the dependent variable vector is 1 more than P. In matrix notation: Yp+l Yp Yp_1 Y1 (p1 8p+1 Yp+2 Yp+| Yp Y2 (p2 8p+2 : + Yt Yt-l Yt-Z Yt—p (pp 8t l l 1 1 Xv = Yp 9p + § 93 Estimating equations are: A _ I ‘l I flip _ (Yp Yp) Yp XP 0 . . : A2 I _| 2M 66 (Yp Yp) Where: 62 : (KP-Ypsep)’ (KP-thpp) T—2p Q: Why 2p in denominator rather than P? A: Because we treat Yl , ..., Yp as presample values and estimate P parameters. Unfortunately, the order P of the AR process is not known in advance. How do we choose P? One way to identify the order of an adequate AR process: - Estimate processes of increasing order K - Test the significance of (pk _. this coefficient is called the kth partial autocorrelation coefficient. 0 It measures the correlation between Y1 and ka not accounted for by an AR(K-l). ° The sequence of partial autocorrelations is called the partial autocorrelation function. - To test the significance of (pk, we need to know the distribution of its estimate (Bk, obtained as the last coordinate of For large sample size, the estimated partial autocorrelations are approximately normally distributed with mean zero and variance l/T, where T is sample size. To check the significance of (pk, form (for example) the 95% confidence intervals: A 2 (pk__: (Pk+'— T 94 Equivalently, check whether the estimated partial autocorrelation falls within the two standard error bounds. Suppose we take the previous univariate series and augment its matrix specification (vertically and horizontally) by another univatiate series (call this new series X,). Motivation: Think of Yt : GDP Xl : Investment Q: Suppose we want to forecast GDP. Given just what was presented on the previous pages, how would we proceed? A: By using the information contained in past values of GDP only. Q: Might not forecasts of GDP be improved if past values of investment are included in the model specification? Recall that GDP = f(investment). A: Yes augment specification horizontally. Q: We know that investment and GDP may influence each other, so why not also use GDP to predict investment: A: This sounds good, so let’s augment specification vertically. Q: Why do all this? A: Admitting this information in this estimation procedure might improve the forecasts. The parallel in econometrics is the movement from a single equation model to a simultaneous equations model. An important issue surfaces as a result of moving in this direction that of causality. Q: Does GDP cause investment, or investment cause GDP, or is it some combination of the two? A: Difficult issue; this will be addressed later. For now, let’s look at the augmented specification without considering causality. Let there be a stochastic process (assumed to be autoregressive) for Y‘ and X,. 95 Yl has been presented and its specification completely laid out. Let’s do the same thing for X. X‘ = (PIXH + ‘szi-z + + (pIXl_p + (0‘ = an AR(P) process Also, (1)l : white noise 13(0)) 2 0 E00505) = 0 for s¢t 1309(0)!) = a: Let there be this “back and forth” influence. Now, develop the vertically and horizontally augmented system. Introduce the following notation on the q): (pij,k where: i = Equation number (= 1 or 2) Equation 1 if left-hand—side variable is Y[ Equation 2 if left—hand-side variable is Xt j = Type of right-hand-side variables included (= 1 or 2) 1 if corresponding right-hand-side variable is a YI 2 if corresponding right-hand—side variable is a Xl k = Order of the process for the respective right-hand-side variable 96 Xt = (P21 lYt-l (9212Yc-2 + + (921,th13 + (P221Xc-1 + (P22,2X:-2 In matrix notation: Yp+l Yp Yp-l Y.1 X X -1 )(1 Y D p Y: 3+2 Y1”1 Yp Y2 )(p+1 xp ' X2 Y z : : 0:. = Y“ YVZ th Xt-l X:—2 Xt—P Xm+l Yp Yp-l Xm+2 0 Yp+l Yp X! a a Yt—l Yt—Z x; l*-< u >4 97 13+! p+2 p‘l 9+2 Characteristics: Notice that the previous model for Y1 is merely augmented by X,. (Notice, we are assuming that Yl is AR(P) and XI AR(M) = AR(P); we have yet to discover the process of each). The grand X matrix is block diagonal. If M = P (as is presently assumed), the two main diagonal blocks are precisely the same. (This has implications for the estimation procedure to use). ° This is a special case of a simultaneous system. Since all the R.H.S. values of X are lagged, each column is predetermined. - The structural form and reduced form coincide. ' Each equation (Yt and X!) can be estimated by OLS. - Notice: 8p+l may be correlated with mp“, 89,2 with (up,2 a, with 03,. Q: Why not use an estimation technique that makes use of the fact that the errors are mutually correlated, e.g., seemingly unrelated regression (SUR) [also known as Zellner estimation]? This could improve the efficiency of the estimators. A: Efficiency improves only if errors are in fact correlated and the respective columns of X are uncorrelated. (This is not the case here.) Furthermore It is not always useful to estimate the autoregressive model in the previous form X=Xfi+a- Q: Why? A: Frequently some of the (pij‘k’s will be zero. Neglecting valid zero restrictions results in inefficient estimates. Notice, for example, that the stochastic process for Yt may very well have different lag structures for both YH and Xt-l than Xt has for Yu and X“. This means that the 2 block diagonal matrices in X are now different. Taking account of the correlation in the error terms now improves the efficiency of the estimators. The variance-covariance matrix of the error term is: 98 and the estimator of Q is I: :2 (XI 25-11) X)_l X, 28(1) X Q: How does one determine an adequate set of regressors for each equation? A: In the univariate case, we based the choice on an adequate order of an AR process on the partial autocorrelations. For multivariate processes the partial autocorrelations are matrices. Hsiao (J ASA 74:553—560) suggests using Akaike’s final prediction error (FPE) criterion; i.e., search for the combination “A” and “B” in each equation for which T + no. of parameters) (3313mm) FPE = (A3) (T — no. of parameters T is a minimum. Notation is as follows: T: sample size used in the estimation. X , XH) are used as regressors in a particular SSEM'B): sum ofsquared errors if YH, ..., Y equation. t-a’ l-I’ Notice the effect of increasing the number of parameters on the right-hand-side of FPE. - It increases the first term. 0 It decreases the second term. It is assumed that these two forces are balanced when their product reaches its minimum. Actual implementation of FPE criterion: - Specify a maximum number (MAX) for A & B that one is willing to consider. ° A univariate AR model is identified for Y| by varying a from zero to MAX. (Suppose that FPEM obtains its minimum at a = a0). ' Choose b between zero and MAX such that b = b0 is the optimum. (Because ao was determined neglecting the x variable in the considered equation, it is possible that 210 is unduly high as a result of an omitted variables effect). Thus, we compute FPE(a,bo) for all a = 0, l, ..., a0 and use the minimizing combination (a,, be). 99 Other methods may also identify adequate numbers oflags ofX and Y. Too little is known about the performance ofthe various methods to attempt a ranking. (Hsiao’s method is introduced here because of its simplicity). Forecasting Consider the previous model in matrix notation. Rearran ge the model so that the parameters appear in the following block form: Yr : (Pm (912.1 Yt—l + + (p||.p (912,]; Yt-p + 81 X! (921,1 (922,1 Xt-l (92”: (p224; Xt-p (01 c1), cpp Y Now, let the one-step ahead forecast at time T be XT” T+l The computation for the one-step ahead forecast is: Y X T T+l-p X = <1)l +...+ q) p T T+l—p 100 ...
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