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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 Ftest results for my unaugmented model and for my augmented model GHJ 698700 Textbook example of conducting the Ftest (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 264272 Introduction to Vector Autoregression (VAR) BM 92 Transition from Enders’ coverage of VAR on pages 264266
and 269270. I have a complete example using two variables:
. Y: Gross Domestic Product (GDP)
. X: Investment BM 93100 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 + B3Yll + [34XlI + 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 + SZYll +Y3XH + (DI
' Notice how the reduced form eliminates any contemporaneous relationship between the left
hand side and the righthand side. . The time—series econometrician calls this a Vector Autoregression system; VAR for short.  The timeseries 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 : (plYtl + ‘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 righthandside 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 Ytl YtZ 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 : (KPYpsep)’ (KPthpp) 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(Kl).
° 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 + ‘sziz + + (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 lefthand—side variable is Y[
Equation 2 if left—handside variable is Xt
j = Type of righthandside variables included (= 1 or 2)
1 if corresponding righthandside variable is a YI
2 if corresponding righthand—side variable is a Xl
k = Order of the process for the respective righthandside variable 96 Xt = (P21 lYtl (9212Yc2 + + (921,th13 + (P221Xc1 + (P22,2X:2
In matrix notation:
Yp+l Yp Ypl 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 Xtl X:—2 Xt—P
Xm+l Yp Ypl
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 Xtl 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 variancecovariance matrix of the error term is: 98 and the estimator of Q is
I: :2 (XI 2511) 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. ta’ lI’ Notice the effect of increasing the number of parameters on the righthandside 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,]; Ytp + 81
X! (921,1 (922,1 Xtl (92”: (p224; Xtp (01
c1), cpp
Y
Now, let the onestep ahead forecast at time T be XT”
T+l The computation for the onestep ahead forecast is: Y
X T T+lp X = <1)l +...+ q)
p T T+l—p 100 ...
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 Fall '11
 BernardMorzuch

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