This chapter’s objectives are to: Describe the theory of stochastic linear difference equations, develop the tools used in estimating ARMA models, consider the time-series properties of stationary and nonstationary models,. | Chapter 2: STATIONARY TIME-SERIES MODELS Applied Econometric Time Series 4th Edition Copyright © 2015 John Wiley & Sons, Inc. STOCHASTIC DIFFERENCE EQUATION MODELS Section 1 Example of a time-series model Although the money supply is a continuous variable, () is a discrete difference equation. Since the forcing process {et} is stochastic, the money supply is stochastic; we can call () a linear stochastic difference equation. If we knew the distribution of {et}, we could calculate the distribution for each element in the {mt} sequence. Since () shows how the realizations of the {mt} sequence are linked across time, we would be able to calculate the various joint probabilities. Notice that the distribution of the money supply sequence is completely determined by the parameters of the difference equation () and the distribution of the {et} sequence. Having observed the first t observations in the {mt} sequence, we can make forecasts of mt+1, mt+2, . () White Noise E(et) = E(et–1) = = 0 E(et)2 = E(et–1) 2 = = s2 [or var(et) = var(et–1) = = s2] E(et et-s) = E(et-j et-j-s) = 0 for all j and s [or cov(et, et-s) = cov(et-j, et-j-s) = 0] A sequence formed in this manner is called a moving average of order q and is denoted by MA(q) 2. ARMA MODELS In the ARMA(p, q) model yt = a0 + a1yt–1 + + apyt-p + et + b1et–1 + + bqet-q where et series are serially uncorrelated “shocks” The particular solution is: Note that all roots must lie outside of the unit circle. If this is the case, we have the MA Representation STATIONARITY Section 3 Stationarity Restrictions for an AR(1) Process Covariance Stationary Series Mean is time-invariant Variance is constant All covariances are constant all autocorrelations are constant Example of a series that are not covariance stationary yt = a + b time yt = yt-1 + et (Random Walk) Formal Definition A stochastic process having a finite mean and variance is covariance stationary if for all t and t s, 1. E(yt) = E(yt-s) = m | Chapter 2: STATIONARY TIME-SERIES MODELS Applied Econometric Time Series 4th Edition Copyright © 2015 John Wiley & Sons, Inc. STOCHASTIC DIFFERENCE EQUATION MODELS Section 1 Example of a time-series model Although the money supply is a continuous variable, () is a discrete difference equation. Since the forcing process {et} is stochastic, the money supply is stochastic; we can call () a linear stochastic difference equation. If we knew the distribution of {et}, we could calculate the distribution for each element in the {mt} sequence. Since () shows how the realizations of the {mt} sequence are linked across time, we would be able to calculate the various joint probabilities. Notice that the distribution of the money supply sequence is completely determined by the parameters of the difference equation () and the distribution of the {et} sequence. Having observed the first t observations in the {mt} sequence, we can make forecasts of mt+1, mt+2, . () White Noise E(et)
