## Archive for July, 2014

The diffusion index forecasts of the CPI (table 3) also represent substantial improvements over the benchmark models. Unlike the IP forecasts, the estimated factors do not account for all of the predictable dynamics in CPI inflation, and adding lags of CPI inflation to the diffusion index forecasts improves their performance, both for fixed к and к selected by recursive BIC. The results for fixed numbers of factors and the autoregressive correction indicate that the MSE attains a minimum at five or six factors. In contrast to the case of IP, the best fixed-к forecast is considerably better than the BIC-selected forecast, in both the balanced and unbalanced panel (the relative RMSEs are .62 v. .71 for the balanced panel with the autoregressive terms, respectively).

Forecasting results. The results of the simulated out of sample forecasting experiments are reported in table 2 for IP and in table 3 for CPI inflation. The entries are the mean squared error (MSE) of the candidate forecasting model, computed relative to the MSE of the autoregressive forecast (so the autoregressive forecast, which is unreported, has a relative MSE of 1.00). Smaller relative MSEs signify more accurate forecasts.

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Phillips curve forecasts. The expectations-augmented Phillips curve constitutes an important tool of empirical macroeconomics and is considered by many to be a reliable tool for forecasting inflation, cf Gordon (1982) and, more recently, the Congressional Budget Office (1996), Fuhrer (1995), Gordon (1997), Staiger, Stock and Watson (1997), and Tootel (1994). For this reason, forecasts based on two variants of a Phillips curve are also included for comparison purposes. These specified the twelve-month inflation rate as the dependent variable:

where pt = ln(CPIt), 7rt = 1200*Apt is monthly CPI inflation at an annual rate, ut is the unemployment rate, and Zt is a vector of variables that control for supply shocks and/or measurement difficulties. The two variants differ in the supply shock variables Zv In one, zt consists solely of the relative price of food and energy; in the other, this relative price is augmented by Gordon’s (1982) variable that controls for the imposition and removal of the Nixon wage and price controls.

For the CPI forecasts, eight leading indicators are used. These variables were chosen because of their good individual performance in previous inflation forecasting exercises. In particular these variables performed well in at least one of the historical episodes considered in Staiger, Stock and Watson (1997).

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Multivariate leading indicator forecasts. The multivariate leading indicator forecasts are of the form,

where {w-t} are various leading indicators that have been used elsewhere to forecast these variables.

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All series on this list were subjected to two preliminary steps: possible transformation by taking logarithms, and possible first differencing. The decision to take logarithms or to first difference the series was judgmentally made. In general, logarithms were taken for all nonnegative series that were not already in rates or percentage units. In general, first differences were taken of real and nominal quantity series and of price indexes. A code summarizing these transformations is given for each series in Appendix B. After these transformations, all series were further standardized to have sample mean zero and unit sample variance.

The factors were estimated using only contemporaneous values of Xt (no stacking of lagged values of Xt). The factors were computed using the algorithms described in section 2. A total of к = 12 factors were estimated.

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For both IP and the CPI, the diffusion index (DI) forecasts were compared to benchmark forecasts from an autoregressive model and multivariate regression-based forecasts using various leading indicators. For the CPI, as an additional comparison forecasts were also computed using models based on the Phillips curve. These various forecasting models are now described in turn.

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Next turn to the results for , which check the consistency predictions of theorem 2. The results for q=r essentially parallel the results for Rj* p although the range of values exceeds the range of R^ pQ. When T = 100 and N=250, is generally large, typically exceeding .95 in the static models. The quality of the forecasts drops in the dynamic models and when there is time variation in the factor loadings.

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The second statistic measures how close the forecast based on the estimated factors is to the infeasible forecast based on the true factors:

According to theorem 2, S~ £ Q 1 either if к = (q+ l)r, or if к > (q+ l)r and the factors included in the forecasting regression are chosen using an information criterion that satisfies condition IC. Accordingly, results are reported for several information criteria: the AIC, the BIC, and the information criterion with the penalty function (3.2) for various choices of the scaling parameter со.

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The initial factor loading matrix Aq is chosen as follows. Let Rj = var( E J=0Xij0Ft-j)/[var( £ 5=0XijOFt-j) + var(eit)l- XijO = MXijO- where XijO is

i.i.d. N(0,1) and independent of {e^, v-t, fyt, vj, and X* is chosen so that R- has a uniform distribution on [0.1,0.8].^ The initial values of the factor Fq are drawn from the stationary distribution of Ft. Finally the {X*t} are transformed to have sample mean zero and sample variance one (this transformation is used in the empirical work presented in the next section).

The scalar variable to be forecast obeys,

The factors were estimated as discussed in section 2 for the balanced panel using {Xjt}, i = 1,… ,N, t = 1,…,T. Estimates were based on the static framework, that is, an augmented X constructed by stacking X and its lags as discussed in section 2.5 was not used. The coefficients f3 in the forecasting regression were estimated by the OLS coefficients $ in the regression of yt_j_ j on £t, t=2,…,T; in particular no lags of were introduced into the forecasting equation. The out-of-sample forecast yj + i was constructed as ут + 1 = fi’Pj. For comparison purposes, the infeasible out-of-sample forecast Уу+1 was also computed, where fP are the OLS coefficients in the regression of yt+ j on F^, t=2,…,T.

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