Lecture 19 - Decomposing a Time Series into its Trend and Cyclical Components It is often assumed that many macroeconomic time series are subject to two sorts of forces: those that influence the long-run behavior of the series and those that influence the short-run behavior of the series. So, for example, growth theory focuses on the forces that influence long-run behavior whereas business cycle theory focuses on the forces that influence short-run behavior. Therefore, it is useful to have econometric methods that can be used to decompose a time series into its trend and cyclical components. [Other reasons – to transform a nonstationary series into a stationary series by removing the trend; alleviate a spurious regression problem.] Several Methods – • Polynomial trend removal • Beveridge-Nelson type decompositions • Hodrick-Prescott FilterI. Polynomial Trend Removal Assume that yt is a trend stationary process. That is: yt = τt + ct τt is a deterministic function, typically a low-order polynomial, of t called the trend (or secular or long-run or permanent) component of yt and ct is a stationary process, called the cyclical component (or short-run or transitory) component of yt. Assume that the trend is a polynomial in t, so that yt = τt + ct = β0 + β1t + … + βptp + ct This is a standard linear regression model with a serially correlated process, ct. How to efficiently estimate the β’s? OLS. (Grenander and Rosenblatt, 1957, show that OLS is asymptotically equivalent to GLS when xt =[1 t … tp].)Then it makes sense to set pptttβββτˆ...ˆˆˆ10+++=and tttycτˆˆ−= where β-hat denotes the OLS estimator. Notes: • In practice p = 1 (linear trend) and, in the case of rapidly growing nominal variables, p = 2 (quadratic trend) tend to be the most common choices of p. • Suppose yt and some or all of the xt’s are increasing (on average) over time. A regression of y on x is likely to be statistically significant even if there is no meaningful or causal relationship between the two. That is, a regression of a trending depending variable on a trending regressor is likely to suggest a statistically significant relationship even when none is present. (What is such a regression called?)In this case it makes some sense to add the explanatory variable t as a separate regressor in the regression: (*) yt = β0 + β1xt + β2t + εt. An alternative: Formulate the model as (**) yt* = β1xt* + εt where yt* is the residual series from the regression of yt on a linear time trend and xt* is the residual series obtained from the regression of xt on a linear trend. Fact: The OLS estimate of β1 obtained from regression (*) and the OLS estimator of β1 obtained from regression (**) will be exactly the same. The residual series will be exactly the same, too. (Frisch-Waugh-Lovell Theorem; see, Greene or Davidson & MacKinnon) [This result extends to regressions with higher order terms in t.]II. Decompositions Based on Differences An alternative to the trend stationary assumption to account for trend behavior in a time series is to assume that the series is difference stationary, i.e., yt is stationary in differenced form.A. Diffence Stationarity A time series yt difference stationary of order d, d a positive integer, if • ∆d yt is stationary • ∆d-1yt is not stationary • The MA form of ∆d yt does not have a “unit root” In practice, d = 1 and, for some rapidly growing nominal time series, d = 2 are the most commonly used values of d. {Suppose yt is the trend stationary process yt = β0 + β1t + εt, where εt is a stationary process. Is yt difference stationary of order one? (Why or why not?)}• The number of differences required to make a time series stationary is also called the “order of integration of the series.” • A stationary series is called an integrated of order zero, I(0), series. A difference stationary series with d = 1 is called an intergrated of order one, I(1), series… • An I(1) process is also called a unit root process because the characteristic polynomial of the AR representation of an I(1) process will have a root equal to 1.B. The Random Walk Process The simplest case of an I(1) process is the random walk: yt = yt-1 + εt , εt a zero-mean iid process Note that for the rw – • ∆yt is an iid process: changes in yt are serially uncorrelated, independent, identically distributed. • dyt+s/dεt = 1 for all s > 0: Innovations have completely permanent effects on the time series! yt+1 = yt + εt+1 = yt+2 = yt+1 + εt+2 = yt-1 + εt +εt+1 + εt+2 … yt+s = yt-1 + (εt +εt+1 + εt+2 + … +