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Econ 240 C. Lecture 3. Part I. Modeling Economic Time Series. Total Returns to Standard and Poors 500, Monthly, 1970-2003. Source: FRED http://research.stlouisfed.org/fred/. Analysis (Decomposition). Lesson one: plot the time series. Model One: Random Walks.

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## Econ 240 C

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**Econ 240 C**Lecture 3**Part I**• Modeling Economic Time Series**Total Returns to Standard and Poors 500, Monthly, 1970-2003**Source: FRED http://research.stlouisfed.org/fred/**Analysis (Decomposition)**• Lesson one: plot the time series**Model One: Random Walks**• Last time we characterized the logarithm of total returns to the Standard and Poors 500 as trend plus a random walk. • Ln S&P 500(t) = trend + random walk = a + b*t + RW(t)**9**8 7 6 5 4 0 100 200 300 400 500 Trace of ln S&P 500(t) Logarithm of Total Returns to Standard & Poors 500 LNSP500 TIME**Analysis(Decomposition)**• Lesson one: Plot the time series • Lesson two: Use logarithmic transformation to linearize**Ln S&P 500(t) = trend + RW(t)**• Trend is an evolutionary process, i.e. depends on time explicitly, a + b*t, rather than being a stationary process, i. e. independent of time • A random walk is also an evolutionary process, as we will see, and hence is not stationary**Model One: Random Walks**• This model of the Standard and Poors 500 is an approximation. As we will see, a random walk could wander off, upward or downward, without limit. • Certainly we do not expect the Standard and Poors to move to zero or into negative territory. So its lower bound is zero, and its model is an approximation.**Model One: Random Walks**• The random walk model as an approximation to economic time series • Stock Indices • Commodity Prices • Exchange Rates**Model Two: White Noise**• Last time we saw that the difference in a random walk was white noise.**Model Two: White Noise**• How good an approximation is the white noise model? • Take first difference of ln S&P 500(t) and plot it and look at its histogram.**The First Difference of ln S&P 500(t)**• D ln S&P 500(t)=ln S&P 500(t) - ln S&P 500(t-1) • D ln S&P 500(t) = a + b*t + RW(t) - {a + b*(t-1) + RW(t-1)} • D ln S&P 500(t) = b + D RW(t) = b + WN(t) • Note that differencing ln S&P 500(t) where both components, trend and the random walk were evolutionary, results in two components, a constant and white noise, that are stationary.**Analysis(Decomposition)**• Lesson one: Plot the time series • Lesson two: Use logarithmic transformation to linearize • Lesson three: Use difference transformation to reduce an evolutionary process to a stationary process**Model Two: White Noise**• Kurtosis or fat tails tend to characterize financial time series**The Lag Operator, Z**• Z x(t) = x(t-1) • Zn x(t) = x(t-n) • RW(t) – RW(t-1) = (1 – Z) RW(t) = D RW(t) = WN(t) • So the difference operator, D, can be written in terms of the lag operator, D= (1 – Z)**Model Three: Autoregressive Time Series of Order One**• An analogy to our model of trend plus shock for the logarithm of the Standard Poors is inertia plus shock for an economic time series such as the ratio of inventory to sales for total business • Source: FRED http://research.stlouisfed.org/fred/**Analogy**• Trend plus random walk: • Ln S&P 500(t) = a + b*t + RW(t) • where RW(t) = RW(t-1) + WN(t) • inertia plus shock • Ratioinvsale(t) = b*Ratioinvsale(t-1) + WN(t)**Model Three: Autoregressive of First Order**• Note: RW(t) = 1*RW(t-1) + WN(t) • where the coefficient b = 1 • Contrast ARONE(t) = b*ARONE(t-1) + WN(t) • What would happen if b were greater than one?**Using Simulation to Explore Time Series Behavior**• Simulating White Noise: • EVIEWS: new workfile, irregular, 1000 observations, GENR WN = NRND**Simulated ARONE Process**• SMPL 1 1, GENR ARONE = WN • SMPL 2 1000 • GENR ARONE =1.1* ARONE(-1) + WN • Smpl 1 1000**First 10 Observations of ARONE**obs WN ARONE 1 -1.204627 -1.204627 2 -1.728779 -3.053869 3 1.478125 -1.881131 4 -0.325830 -2.395073 5 -0.593882 -3.228463 6 0.787438 -2.763872 7 0.157040 -2.883219 8 -0.211357 -3.382898 9 -0.722152 -4.443340 10 0.775963 -4.111711**Model Three: Autoregressive**• What if b= -1.1? • ARONE*(t) = -1.1*ARONE*(t-1) + WN(t) • SMPL 1 1, GENR ARONE* = WN • SMPL 2 1000 • GENR ARONE* = -1.1*ARONE*(-1) + WN • SMPL 1 1000**Model Three: Conclusion**• For Stability ( stationarity) -1<b<1**Part II**• Forecasting: A preview of coming attractions**Ratio of Inventory to Sales**• EVIEWS Model: Ratioinvsale(t) = c + AR(1) • Ratioinvsale is a constant plus an autoregressive process of the first order • AR(t) = b*AR(t-1) + WN(t) • Note: Ratioinvsale(t) - c = AR(t), so • Ratioinvsale(t) - c = b*{ Ratioinvsale(t-1) - c} + WN (t)**Ratio of Inventory to Sales**• Use EVIEWS to estimate coefficients c and b. • Forecast of Ratioinvsale at time t is based on knowledge at time t-1 and earlier (information base) • Forecast at time t-1 of Ratioinsale at time t is our expected value of Ratioinvsale at time t**One Period Ahead Forecast**• Et-1[Ratioinvsale(t)] is: • Et-1[Ratioinvsale(t) - c] = • Et-1[Ratioinvsale(t)] - c = • Forecast - c = b*Et-1[Ratioinvsale(t-1) - c] + Et-1[WN(t)] • Forecast = c + b*Ratioinvsale(t-1) -b*c + 0**Dependent Variable: RATIOINVSALE**Method: Least Squares Date: 04/08/03 Time: 13:56 Sample(adjusted): 1992:02 2003:01 Included observations: 132 after adjusting endpoints Convergence achieved after 3 iterations Variable Coefficient Std. Error t-Statistic Prob. C 1.417293 0.030431 46.57405 0.0000 AR(1) 0.954517 0.024017 39.74276 0.0000 R-squared 0.923954 Mean dependent var 1.449091 Adjusted R-squared 0.923369 S.D. dependent var 0.046879 S.E. of regression 0.012977 Akaike info criterion -5.836210 Sum squared resid 0.021893 Schwarz criterion -5.792531 Log likelihood 387.1898 F-statistic 1579.487 Durbin-Watson stat 2.674982 Prob(F-statistic) 0.000000 Inverted AR Roots .95**Forecast for Ratio of Inventory to Sales for February 2003**• E2003:01 [Ratioinvsale(2003:02)= c - b*c + b*Ratioinvsale(2003:02) • Forecast = 1.417 - 0.954*1.417 + 0.954*1.360 • Forecast = 0.06514 + 1.29744 • Forecast = 1.36528**How Well Do We Know This Value of the Forecast?**• Standard error of the regression = 0.0130 • Approximate 95% confidence interval for the one period ahead forecast = forecast +/- 2*SER • Ratioinvsale(2003:02) = 1.36528 +/- 2*.0130 • interval for the forecast 1.34<forecast<1.39**Lessons About ARIMA Forecasting Models**• Use the past to forecast the future • “sophisticated” extrapolation models • competitive extrapolation models • use the mean as a forecast for a stationary time series, Et-1[y(t)] = mean of y(t) • next period is the same as this period for a stationary time series and for random walks, Et-1[y(t)] = y(t-1) • extrapolate trend for an evolutionary trended time series, Et-1[y(t)] = a + b*t = y(t-1) + b

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