One way to estimate a distributed lag model is simply to include all Nx lags of x in the regression, which can be estimated by least squares in the usual way. Such polynomially distributed delays promote smoothness in the delay distribution and can lead to sophisticated simple models with improved forecasting performance. Polynomial distributed lag models are estimated by minimizing the sum of squared residuals in the usual way, with the restriction that the lag weights follow a low-order polynomial whose degree must be specified.
An alternative and often preferred approach uses rational distributed lags, which we introduced in Chapter 7 in the context of univariate modeling ARM A. Each variable is related not only to its own past, but also to the past of all other variables in the system. In the unconstrained V ARs we have studied so far, everything causes everything else because the lags of each variable appear on the right-hand side of each equation.
To understand what predictive causality means in the context of aV AR(p), consider the jth equation of the N equation system, which has yj on the left and p lags of each of the N variables on the right. If yi causes yj, then at least one of the lags of yi appearing on the right-hand side of the yj equation must have a nonzero coefficient. In the bivariate V AR, this implies non-causality in terms of h-step-ahead forecast errors, for alh.
And we are often very interested in the answers to questions like "Does yi contribute to getting better.
Impulse-Response Functions
Note carefully that b0 gives the immediate effect of the shock at time t, when it hits. The parameter b1, which multiplies ε0t−1, represents the effect of the shock a period later, and so on. The full set of impulse response coefficients, {b0, b1, ..}, follows the full dynamic response of y to the shock.
As in the one-sided case, it is fruitful to adopt a different normalization of the moving average representation for impulse response analysis. The multivariate analogue of our univariate normalization with σ is called normalization by the Cholesky factor.5 The resulting moving average representation The VAR has a number of useful features that exactly parallel the univariate case. It often turns out that impulse response functions are not sensitive to order, but the only way to be sure is to check.7.
In practical applications of impulse response analysis, we simply replace unknown parameters by estimates, which immediately yield point estimates of the impulse response functions. However, it is more difficult to obtain confidence intervals for impulse response functions, and adequate procedures are still under development.
Variance Decompositions
This is crucial because it allows us to perform the experiment of interest—shocking one variable separately from the others, which we can do if the innovations are uncorrelated but cannot if they are correlated, as in the original unnormalized representation. After normalizing the system for a given order, say first y1, we calculate four sets of impulse response functions for the bivariate model: response y1 to unit normalized innovation on y1, { b011, b111, b211,.
Application: Housing Starts and Completions
The patterns in the sample autocorrelations and partial autocorrelations are highly statistically significant, as evidenced by both the Bartlett standard errors and the Ljung-Box Q statistics. Starts and completions are highly correlated across all displacements, and a clear pattern also emerges: although the contemporaneous correlation is high (.78), completions are maximally correlated with starts delayed by about 6-12 months (about .90 ). Again, this makes good sense given the time it takes to build a house.
The explanatory power of the model is good, as assessed by R2, as well as by plots of actual and fitted values, and the residuals appear white, as assessed by residual sample autocorrelations, partial autocorrelations, and the Ljung-Box statistic. Also note that neither lag of completions has a significant effect on starts, which makes sense - we obviously expect starts to cause completions, not the other way around. The hypothesis that starts do not cause completions is simply that all coefficients at the four lags of starts in the completions equation are zero.
To get an idea of the dynamics of the estimated VAR before making predictions, we calculate impulse-response functions and variance decompositions. We present the results for starts first in the order, so that a current innovation to start affects only the current starts, but the results are robust to reversing the order. The fraction of error variance in predicting takeoffs due to innovations in takeoffs is almost 100 percent at all horizons.
In contrast, the fraction of the error variance in forecast completions due to innovations in startups is nearly zero at short horizons, but rises steadily and is nearly 100 percent at long horizons, again reflecting time-to-build effects. The figure also makes clear that the housing recovery from the 1990 recession was slower than the previous recovery in the sample, which of course makes forecasting difficult.
Exercises, Problems and Complements
Even in the early 1970s, time series analysis was mostly univariate and made little use of economic theory. As Klein (1981) notes, however, complex econometric system estimation methods have had little value in practical forecasting and have therefore been largely abandoned, while rationally distributed lag patterns associated with time series models have led to large improvements in practical forecasting accuracy. 9 Thus, in recent times the distinction between econometrics and time series analysis has largely disappeared, as the union has embraced the best of both. In many ways, V AR is a modern embodiment of both the econometric and time series traditions.
8Klein and Young (1980) and Klein (1983) provide good discussions of the traditional econometric paradigm of simultaneous equations, as well as the relationship between structural simultaneous equation models and reduced-form time series models. Wallis (1995) provides a good summary of modern large-scale macroeconomic modeling and forecasting, and Pagan and Robertson (2002) provides an intriguing discussion of the variety of macroeconomic forecasting approaches currently used by central banks around the world . Business cycles are a type of fluctuations that occur in the overall economic activity of countries that organize their work mainly in business: a cycle consists of expansions that occur at approximately the same time in many economic activities, followed by similar general recessions. contractions and revivals that transition into the expansion phase of the next cycle.
Burns and Mitchell used the clusters of turning points in individual series to determine the monthly dates of the turning points in the overall business cycle, and to construct composite indexes of leading, coincident, and lagging indicators. Such indexes have been produced by the National Bureau of Economic Research (a think tank in Cambridge, Massachusetts), the Department of Commerce (a U.S. government agency in Washington, DC), and the Conference Board (a business membership organization based in New York). York).11 Composite indexes of leading indicators are often used to measure likely future economic developments, but their usefulness is certainly not without controversy and remains the subject of ongoing research. 11The indexes build on very early work, such as Harvard's 'Index of General Business Condition'. For a fascinating discussion of the early work, see Hardy (1923), chapter 7.
The study of the phenomenon dates to the early twentieth century, and a key study by Granger and Newbold (1974) showed the prevalence and potential seriousness of the problem. Using the house start and finish data on the book's website, you can compare the forecasting performance of the VAR used in this chapter with that of its obvious competitor: univariate autoregression. Now extend the model to capture cross-variable dynamics using a rationally distributed layer of the second variable, yielding the general transfer.
12Table 1 shows a range of important forecasting models, all of which are special cases of the transfer function model. Distributed lag regression with lagged dependent variables is a potentially restrictive special case that occurs when C(L) = 1 and B(L) = D(L). For example, the univariate time series used is often IM A(0,1,1) (ie, exponential smoothing or local level), which Hendry, Clements, and others have argued is robust to shifts.
Notes
