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Ka-fu Wong University of Hong Kong

Ka-fu Wong University of Hong Kong. Forecasting with Regression Models. Linear regression models. Endogenous variable. Exogenous variables. Explanatory variables. Rule , rather than exception: all variables are endogenous. Conditional forecasting.

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Ka-fu Wong University of Hong Kong

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  1. Ka-fu WongUniversity of Hong Kong Forecasting with Regression Models

  2. Linear regression models Endogenous variable Exogenous variables Explanatory variables Rule, rather than exception: all variables are endogenous.

  3. Conditional forecasting The h-step ahead forecast of y given some assumed h-step-ahead value of xT+h. Assumed h-step-ahead value of the exogenous variables Call it scenario analysis or contingency analysis – based on some assumed h-step-ahead value of the exogenous variables.

  4. Uncertainty of Forecast • Specification uncertainty / error: our models are only approximation (since no one knows the truth). E.g., we adopt an AR(1) model but the truth is AR(2). • Almost impossible to account for via a forecast interval. • Parameter uncertainty / sampling error: parameters are estimated from a data sample. The estimate will always be different from the truth. The difference is called sampling error. • Can account for via a forecast interval if we do the calculation carefully. • Innovation uncertainty: errors that cannot be avoided even if we know the true model and true parameter. This is, unavoidable. • Often account for via a forecast interval using standard softwares.

  5. Quantifying the innovation and parameter uncertainty Consider the very simple case in which x has a zero mean:

  6. Density forecast that accounts for parameter uncertainty ~

  7. Interval forecasts that do not acknowledge parameter uncertainty

  8. Interval forecasts that do acknowledge parameter uncertainty The closer xT+h* is closer to its mean, the smaller is the prediction-error variance.

  9. Unconditional Forecasting Models Forecast based on some other models of x, say, by assuming x to follow an AR(1).

  10. h-step-ahead forecast without modeling x explicitlyBased on unconditional forecasting models • Standing at time T, with observations, (x1,y1), (x2,y2),…,(xT,yT) • 1-step-ahead: • yt = b0 + b1 xt-1 + et • yT+1 = b0 + b1 xT + et • 2-step-ahead: • yt = b0 + b1 xt-2 + et • yT+2 = b0 + b1 xT + et • … • h-step-ahead: • yt = b0 + b1 xt-h + et • yT+h = b0 + b1 xT + et

  11. h-step-ahead forecast without modeling x explicitlyBased on unconditional forecasting models • Special cases: • The model contains only time trends and seasonal components. • Because these components are perfectly predictable.

  12. Distributed Lags y depends on a distributed lags of past x’s b0, d1,…,dNx Parameters to be estimated:

  13. Polynomial Distributed Lags b0, a, b, c Parameters to be estimated:

  14. Rational Distributed Lags Example: A(L) = a0 + a1L B(L) = b0 + b1L b0 yt + b1 yt-1 = a0 xt + a1 xt-1 + b0et + b1et-1 yt = [- b1 yt-1 + a0xt + a1xt-1 + b0et + b1et-1]/b0 yt = [- b1/b0] yt-1 + [a0/b0] xt + [a1/b0] xt-1 + et + [b1/b0] et-1

  15. Regression model with AR(1) disturbance

  16. ARMA(p,q) models equivalent to model with only a constant regressor and ARMA(p,q) disturbances.

  17. Transfer function models A transfer function is a mathematical representation of the relation between the input and output of a system.

  18. Vector Autoregressions, VAR(p)allows cross-variable dynamics VAR(1) of two variables. The variable vector consists of two elements. The innovations allowed to be correlated. Regressors consist of the variable vector lagged one period only.

  19. Estimation of Vector Autoregressions Run OLS regressions equation by equation. OLS estimation turns out to have very good statistical properties when each equation has the same regressors, as in standard VARs. Otherwise, a more complicated estimation procedure called seemingly unrelated regression, which explicitly accounts for correlation across equation disturbances, would be need to obtain estimates with good statistical properties.

  20. The choice order Estimation of Vector Autoregressions Use AIC and SIC.

  21. Forecast Estimation of Vector Autoregressions Given the parameters, or parameter estimates y1,T, y2,T y1,T+1, Y2,T+1 y1,T+1, y2,T+1 y1,T+2, Y2,T+2 y1,T+2, y2,T+2 y1,T+3, Y2,T+3 y1,T+3, y2,T+3

  22. Predictive Causality • Two principles • Cause should occur before effect. • A causal series should contain information useful for forecasting that is not available in the other series. • Predictive Causality in a VAR y2 does not cause y1 if φ12 =0 In a bivariate VAR, noncausality in 1-step-ahead forecast will imply noncausality in h-step-ahead forecast.

  23. Predictive Causality • In VAR with higher dimension, noncausality in 1-step-ahead forecast need not imply noncausality in h-step-ahead forecast. • Example: • Variable i may 1-step-cause variable j • Variable j may 1-step-cause variable k • Variable i 2-step-causes variable k but does not 1-step-cause variable k.

  24. Impulse response functions All univariate ARMA(p,q) processes can be written as: We can always normalize the innovations with a constant m:

  25. Impulse response functions Impact of et on yt: 1 unit increase in et’ is equivalent to one standard deviation increase in et. 1 unit increase in et’ has b0’ impact on yt 1 standard deviation increase in et has b0s impact on yt, b1s impact on yt, etc.

  26. AR(1)

  27. VAR(1)

  28. Normalizing the VAR by the Cholesky factor If y1 is ordered first, Example: y1 = GDP, y2 = Price level An innovation to GDP has effects on current GDP and price level. An innovation to price level has effects only on current price level but not current GDP.

  29. Features of Cholesky decomposition • The innovations of the transformed system are in standard deviation units. • The current innovations in the normalized representation have can non-unit coefficients. • The first equation has only one current innovation, e1,t. The second equation has both current innovations. • The normalization yields a zero covariance between the innovations.

  30. Normalizing the VAR by the Cholesky factor If y2 is ordered first, Example: y1 = GDP, y2 = Price level An innovation to price level has effects on current GDP and price level. An innovation to GDP has effects only on current GDP but not current price level.

  31. Impulse response functions • With bivariate autoregression, we can compute four sets of impulse-response functions: • y1 innovations (e1,t) on y1 • y1 innovations (e1,t) on y2 • y2 innovations (e2,t) on y1 • y2 innovations (e2,t) on y2

  32. Variance decomposition • How much of the h-step-ahead forecast error variance of variable i is explained by innovations to variable j, for h=1,2,…. ? • With bivariate autoregression, we can compute four sets of variance decomposition: • y1 innovations (e1,t) on y1 • y1 innovations (e1,t) on y2 • y2 innovations (e2,t) on y1 • y2 innovations (e2,t) on y2

  33. Example:y1 = Housing starts, y2= Housing completions (1968:01 – 1996:06) group fig112 starts comps freeze(Figure112) fig112.line(d) Observation #1: Seasonal pattern. Observation #2: Highly cyclical with business cycles. Observation #3: Completions lag starts.

  34. Correlogram and Ljung-Box Statistics of housing starts (1968:01 to 1991:12) freeze(Table112) starts.correl(24)

  35. Correlogram and Ljung-Box Statistics of housing starts (1968:01 to 1991:12)

  36. Correlogram and Ljung-Box Statistics of housing completions (1968:01 to 1991:12) freeze(Table113) comps.correl(24)

  37. Correlogram and Ljung-Box Statistics of housing starts (1968:01 to 1991:12)

  38. Starts and completions, sample cross-correlations freeze(Figure115) fig112.cross(24) starts comps

  39. VAR regression by OLS (1) equation Table114.ls starts c starts(-1) starts(-2) starts(-3) starts(-4) comps(-1) comps(-2) comps(-3) comps(-4)

  40. VAR regression by OLS (1)

  41. VAR regression by OLS (1)

  42. VAR regression by OLS (1)

  43. VAR regression by OLS (2) equation Table116.ls comps c starts(-1) starts(-2) starts(-3) starts(-4) comps(-1) comps(-2) comps(-3) comps(-4)

  44. VAR regression by OLS (2)

  45. VAR regression by OLS (2)

  46. VAR regression by OLS (2)

  47. Predictive causality test group tbl108 comps starts freeze(Table118) tbl108.cause(4)

  48. Impulse response functions(response to one standard-deviation innovations) var fig1110.ls 1 4 starts comps freeze(Figure1110) fig1110.impulse(36,m)

  49. Variance decomposition freeze(Figure1111) fig1110.decomp(36,m)

  50. Starts: History, 1968:01-1991:12Forecast, 1992:01-1996:06

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