Noncausal Autoregressions for Economic Time Series

Abstract

This paper is concerned with univariate noncausal autoregressive models and their potential usefulness in economic applications. In these models, future errors are predictable, indicating that they can be used to empirically approach rational expectations models with nonfundamental solutions. In the previous theoretical literature, nonfundamental solutions have typically been represented by noninvertible moving average models. However, noncausal autoregressive and noninvertible moving average models closely approximate each other, and therefore,the former provide a viable and practically convenient alternative. We show how the parameters of a noncausal autoregressive model can be estimated by the method of maximum likelihood and derive related test procedures. Because noncausal autoregressive models cannot be distinguished from conventional causal autoregressive models by second order properties or Gaussian likelihood, a model selection procedure is proposed. As an empirical application, we consider modeling the U.S. in ‡ation which, according to our results, exhibits purely forward-looking dynamics.

Autoregression-based estimation of the new Keynesian Phillips curve

“…The starting point of our procedure for estimating the NKPC is an adequate noncausal AR model for in ‡ation, and in this section, we brie ‡y describe the noncausal AR model of Lanne and Saikkonen (2011a). Consider a stochastic process y t (t = 0; 1; 2; :::) generated by…”

“…The function f (x; !) is assumed to satisfy the regularity conditions stated in Andrews et al (2006) and Lanne and Saikkonen (2011a). These conditions imply that f (x; !)…”

“…For the U.S. in ‡ation we use Student's t distribution as the error distribution in Section 4. Lanne and Saikkonen (2011a) showed how model (1) can be consistently estimated by the method of maximum likelihood (ML). They also showed that the (local) ML estimator is asymptotically normally distributed, and a consistent estimator of the limiting covariance matrix is obtained in the usual way from the standardized Hessian of the log-likelihood function.…”

“…Moreover, in this case, endogeneity of such an instrument is not reliably revealed by Hansen's (1982) J test. Noncausality of in ‡ation found by Lanne and Saikkonen (2011a) and Lanne et al (2011) thus indicates that using lags of in ‡ation as instruments as is commonly done in the previous literature, is likely to yield misleading results. Lanne and Saikkonen (2011b) also found noncausality very common in a comprehensive data set compirising more than 300 macroeconomic and …nancial time series, which suggests that …nding valid additional instruments for the estimation of the NKPC 2 may be challenging.…”

“…Section 2 describes the noncausal AR model of Lanne and Saikkonen (2011a) and discusses model selection. In Section 3, we derive the maximum likelihood estimator of the NKPC based on the selected noncausal AR model for in ‡ation.…”