Econ 240C Lecture 18

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

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Review 2. Ideas That Are Transcending

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

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 b1 = 0, then e1 = edcapu

Standard VAR (lecture 17) dcapu(t) = (a1 + b1 a2)/(1- b1 b2) +[ (g11 + b1 g21)/(1- b1 b2)] dcapu(t-1) + [ (g12 + b1 g22)/(1- b1 b2)] dffr(t-1) + [(d1 + b1 d2 )/(1- b1 b2)] x(t) + (edcapu (t) + b1 edffr (t))/(1- b1 b2) But if we assume b1 =0, then dcapu(t) = a1 +g11 dcapu(t-1) + g12 dffr(t-1) + d1 x(t) + edcapu (t) +

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

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

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

Review 2. Ideas That Are Transcending

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

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

Lab 7 240 C

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

Lab 8 240 C

Model with no Intervention Variable

Add seasonal difference of differenced step function

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

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

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

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

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)

Lecture 3, 240 C: Trace of ln S&P 500(t) Logarithm of Total Returns to Standard & Poors 500 4 5 6 7 8 9 100 200 300 400 500 LNSP500 TIME

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.

Trace of ln S&P 500(t) – ln S&P(t-1)

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

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 PR(t)/ln[PG(t) + F(t)] = diff(t) Know cointegrating equation: 1* ln PR(t) – 1* ln[PG(t) + F(t)] = diff(t) So do a unit root test on diff, which should be stationary; check with Johansen test

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

Autoregressive Models AR(t) = b1 AR(t-1) + b2 AR(t-2) + …. + bp AR(t-p) + WN(t) AR(t) - b1 AR(t-1) - b2 AR(t-2) - …. + bp AR(t-p) = WN(t) [1 - b1 Z + b2 Z2 + …. bp Zp ] AR(t) = WN(t) B(Z) AR(t) = WN(t) AR(t) = [1/B(Z)]*WN(t) AR(t) 1/B(Z) WN(t)

Moving Average Models MA(t) = WN(t) + a1 WN(t-1) + a2 WN(t-2) + …. aq WN(t-q) MA(t) = WN(t) + a1 Z WN(t) + a2 Z2 WN(t) + …. aq Zq WN(t) MA(t) = [1 + a1 Z + a2 Z2 + …. aq Zq ] WN(t) MA(t) = A(Z)*WN(t) MA(t) A(Z) WN(t)

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

Distributed Lag Models y(t) = h0 x(t) + h1 x(t-1) + …. hn x(t-n) + resid(t) y(t) = h0 x(t) + h1 Zx(t) + …. hn Zn x(t) + resid(t) y(t) = [h0 + h1 Z + …. hn Zn ] x(t) + resid(t) y(t) = h(Z)*x(t) + resid(t) note x(t) = Ax (Z)/Bx (Z) WNx (t), or [Bx (Z) /Ax (Z)]* x(t) =WNx (t), so [Bx (Z) /Ax (Z)]* y(t) = h(Z)* [Bx (Z) /Ax (Z)]* x(t) + [Bx (Z) /Ax (Z)]* resid(t) or W(t) = h(Z)*WNx (t) + Resid*(t)

Distributed Lag Models Where w(t) = [Bx (Z) /Ax (Z)]* y(t) and resid*(t) = [Bx (Z) /Ax (Z)]* resid(t) cross-correlation of the orthogonal WNx (t) with w(t) will reveal the number of lags n in h(Z), and the signs of the parameters h0 , h1 , etc. for modeling the regression of w(t) on a distributed lag of the residual, WNx (t), from the ARMA model for x(t)

Economic Models of Time Series Interest Rate Parity

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

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

Historical Interest Rates & Historical Exchange Rates Dollar Interest spread Yen

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