Econ 240C Lecture 18

2 Review 2002 Final Ideas that are transcending p. 15 Economic Models of Time Series Symbolic Summary

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15 Review 2. Ideas That Are Transcending

16 Use the Past to Predict the Future A. Applications Trend Analysis –linear trend –quadratic trend –exponential trend ARIMA Models –autoregressive models –moving average models –autoregressive moving average models

17 Use Assumptions To Cope With Constraints A. Applications 1. Limited number of observations: simple exponential smoothing –assume the model: (p, d, q) = (0, 1, 1) 2. No or insufficient identifying exogenous variables: interpreting VAR impulse response functions –assume the error structure is dominated by one pure error or the other, e.g assume   = 0, then e 1 = e dcapu

18 Standard VAR (lecture 17) dcapu(t) = (        /(1-     ) +[ (    +     )/(1-     )] dcapu(t-1) + [ (    +     )/(1-     )] dffr(t-1) + [(    +      (1-     )] x(t) + (e dcapu  (t) +   e dffr  (t))/(1-     ) But if we assume    then  dcapu(t) =    +    dcapu(t-1) +   dffr(t-1) +    x(t) + e dcapu  (t) + 

19 Use Assumptions To Cope With Constraints A. Applications 3. No or insufficient identifying exogenous variables: simultaneous equations –assume the error structure is dominated by one error or the other, tracing out the other curve

20 Simultaneity There are two relations that show the dependence of price on quantity and vice versa –demand: p = a - b*q +c*y + e p –supply: q= d + e*p + f*w + e q

21 demand price quantity Shift in demand with increased income, may trace out i.e. identify or reveal the demand curve supply

22 Review 2. Ideas That Are Transcending

23 Reduce the unexplained sum of squares to increase the significance of results A. Applications 1. 2-way ANOVA: using randomized block design –example: minutes of rock music listened to on the radio by teenagers Lecture 1 Notes, 240 C we are interested in the variation from day to day to get better results, we control for variation across teenager

27 Reduce the unexplained sum of squares to increase the significance of results A. Applications 2. Distributed lag models: model dependence of y(t) on a distributed lag of x(t) and –model the residual using ARMA

28 Lab C

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30 Reduce the unexplained sum of squares to increase the significance of results A. Applications 3. Intervention Models: model known changes (policy, legal etc.) by using dummy variables, e.g. a step function or pulse function

31 Lab C

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33 Model with no Intervention Variable

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35 Add seasonal difference of differenced step function

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37 Review 2002 Final Ideas that are transcending Economic Models of Time Series Symbolic Summary

38 Time Series Models Predicting the long run: trend models Predicting short run: ARIMA models Can combine trend and arima –Differenced series Non-stationary time series models –Andrew Harvey “structural models using updating and the Kalman filter –Artificial neural networks

39 The Magic of Box and Jenkins Past patterns of time series behavior can be captured by weighted averages of current and lagged white noise: ARIMA models Modifications (add-ons) to this structure –Distributed lag models –Intervention models –Exponential smoothing –ARCH-GARCH

40 Economic Models of Time Series Total return to Standard and Poors 500

41 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)

42 Lecture 3, 240 C: Trace of ln S&P 500(t) TIME LNSP500 Logarithm of Total Returns to Standard & Poors 500

43 The First Difference of ln S&P 500(t)  ln S&P 500(t)=ln S&P 500(t) - ln S&P 500(t-1)  ln S&P 500(t) = a + b*t + RW(t) - {a + b*(t-1) + RW(t-1)}  ln S&P 500(t) = b +  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.

44 Trace of ln S&P 500(t) – ln S&P(t-1)

45 Histogram of ln S&P 500(t) – ln S&P(t-1)

46 Cointegration Example The Law of One Price Dark Northern Spring wheat –Rotterdam import price, CIF, has a unit root –Gulf export price, fob, has a unit root –Freight rate ambiguous, has a unit root at 1% level, not at the 5% Ln P R (t)/ln[P G (t) + F(t)] = diff(t) –Know cointegrating equation: –1* ln P R (t) – 1* ln[P G (t) + F(t)] = diff(t) –So do a unit root test on diff, which should be stationary; check with Johansen test

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54 Review 2002 Final Ideas that are transcending Economic Models of Time Series Symbolic Summary

55 Autoregressive Models AR(t) = b 1 AR(t-1) + b 2 AR(t-2) + …. + b p AR(t-p) + WN(t) AR(t) - b 1 AR(t-1) - b 2 AR(t-2) - …. + b p AR(t-p) = WN(t) [1 - b 1 Z + b 2 Z 2 + …. b p Z p ] AR(t) = WN(t) B(Z) AR(t) = WN(t) AR(t) = [1/B(Z)]*WN(t) WN(t)1/B(Z)AR(t)

56 Moving Average Models MA(t) = WN(t) + a 1 WN(t-1) + a 2 WN(t-2) + …. a q WN(t-q) MA(t) = WN(t) + a 1 Z WN(t) + a 2 Z 2 WN(t) + …. a q Z q WN(t) MA(t) = [1 + a 1 Z + a 2 Z 2 + …. a q Z q ] WN(t) MA(t) = A(Z)*WN(t) WN(t)A(Z)MA(t)

57 ARMA Models ARMA(p,q) = [A q (Z)/B p (Z)]*WN(t) WN(t)A(Z)/B(Z)ARMA(t)

58 Distributed Lag Models y(t) = h 0 x(t) + h 1 x(t-1) + …. h n x(t-n) + resid(t) y(t) = h 0 x(t) + h 1 Zx(t) + …. h n Z n x(t) + resid(t) y(t) = [h 0 + h 1 Z + …. h n Z n ] x(t) + resid(t) y(t) = h(Z)*x(t) + resid(t) note x(t) = A x (Z)/B x (Z) WN x (t), or [B x (Z) /A x (Z)]* x(t) =WN x (t), so [B x (Z) /A x (Z)]* y(t) = h(Z)* [B x (Z) /A x (Z)]* x(t) + [B x (Z) /A x (Z)]* resid(t) or W(t) = h(Z)*WN x (t) + Resid*(t)

59 Distributed Lag Models Where w(t) = [B x (Z) /A x (Z)]* y(t) and resid*(t) = [B x (Z) /A x (Z)]* resid(t) cross-correlation of the orthogonal WN x (t) with w(t) will reveal the number of lags n in h(Z), and the signs of the parameters h 0, h 1, etc. for modeling the regression of w(t) on a distributed lag of the residual, WN x (t), from the ARMA model for x(t)

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61 Economic Models of Time Series Interest Rate Parity

62 How is exchange rate determined? The Asset Approach – based upon “interest rate parity” Monetary Approach – based upon “purchasing power parity” The key element > Expected Rate of Return Investors care about Real rate of return Risk Liquidity

63 The basic equilibrium condition in the foreign exchange market is interest parity. Uncovered interest parity R $ =R ¥ +(E e $/¥ -E $/¥ )/E $/¥ -Risk Premium Covered interest rate parity (risk-free) R $ =R ¥ +(F $/¥ -E $/¥ )/E $/¥

64 Historical Interest Rates & Historical Exchange Rates Dolla r Yen Interes t spread

65 Explaining the Spread (Dollar vs. Yen) Interest spread Change in Exchange rate Interest parity