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This chapter’s objectives are to: Explain how stochastic difference equations can be used for forecasting and illustrate how such equations can arise from familiar economic models, explain what it means to solve a difference equation, demonstrate how to find the solution to a stochastic difference equation using the iterative method,. | Chapter 1: Difference Equations Applied Econometric Time Series Fourth Edition 1 TIME-SERIES MODELS Section 1 The traditional use of time series models was for forecasting If we know yt+1 = a0 + a1yt + et+1 then Etyt+1 = a0 + a1yt and since yt+2 = a0 + a1yt+1 + et+2 Etyt+2 = a0 + a1Etyt+1 = a0 + a1(a0 + a1yt) = a0 + a1a0 + (a1)2yt 3 Capturing Dynamic Relationships With the advent of modern dynamic economic models, the newer uses of time series models involve Capturing dynamic economic relationships Hypothesis testing Developing “stylized facts” In a sense, this reverses the so-called scientific method in that modeling goes from developing models that follow from the data. 4 The Random Walk Hypothesis yt+1 = yt + et+1 or Dyt+1 = et+1 where yt = the price of a share of stock on day t, and et+1 = a random disturbance term that has an expected value of zero. Now consider the more general stochastic difference equation Dyt+1 = a0 + a1yt + et+1 The random walk hypothesis requires the testable restriction: a0 = a1 = 0. 5 The Unbiased Forward Rate (UFR) hypothesis Given the UFR hypothesis, the forward/spot exchange rate relationship is: st+1 = ft + et+1 (1.6) where et+1 has a mean value of zero from the perspective of time period t. Consider the regression st+1 = a0 + a1ft + t+1 The hypothesis requires a0 = 0, a1 = 1, and that the regression residuals t+1 have a mean value of zero from the perspective of time period t. The spot and forward markets are said to be in long-run equilibrium when et+1 = 0. Whenever st+1 turns out to differ from ft, some sort of adjustment must occur to restore the equilibrium in the subsequent period. Consider the adjustment process st+2 = st+1 – a[ st+1 – ft ] + est+2 a > 0 (1.7) ft+1 = ft + b [ st+1 – ft ] + eft+1 b > 0 (1.8) where est+2 andeeft+1 both have an expected value of zero. 6 Trend-Cycle Relationships We can think of a time series as being composed of: yt = trend + “cycle” + noise Trend: Permanent Cycle: predictable (albeit . | Chapter 1: Difference Equations Applied Econometric Time Series Fourth Edition 1 TIME-SERIES MODELS Section 1 The traditional use of time series models was for forecasting If we know yt+1 = a0 + a1yt + et+1 then Etyt+1 = a0 + a1yt and since yt+2 = a0 + a1yt+1 + et+2 Etyt+2 = a0 + a1Etyt+1 = a0 + a1(a0 + a1yt) = a0 + a1a0 + (a1)2yt 3 Capturing Dynamic Relationships With the advent of modern dynamic economic models, the newer uses of time series models involve Capturing dynamic economic relationships Hypothesis testing Developing “stylized facts” In a sense, this reverses the so-called scientific method in that modeling goes from developing models that follow from the data. 4 The Random Walk Hypothesis yt+1 = yt + et+1 or Dyt+1 = et+1 where yt = the price of a share of stock on day t, and et+1 = a random disturbance term that has an expected value of zero. Now consider the more general stochastic difference equation Dyt+1 = a0 + a1yt + et+1 The random walk hypothesis requires the .
