1. Econ 240C
Lecture 18

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

5. 2

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 b1 = 0, then e1 = edcapu

18. 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) +

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 + ep
supply: q= d + e*p + f*w + eq

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

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

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

33. Model with no Intervention Variable

35. Add seasonal difference of differenced step function

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)
Logarithm of Total Returns to Standard & Poors 500 4 5 6 7 8 9
100
200
300
400
500
LNSP500
TIME

43. 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.

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 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

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

55. 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)

56. 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)

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

58. 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)

59. 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)

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$=R¥+(Ee$/¥-E$/¥)/E$/¥-Risk Premium
Covered interest rate parity (risk-free)
R$=R¥+(F$/¥-E$/¥)/E$/¥

64. Historical Interest Rates & Historical Exchange Rates
Dollar
Interest spread
Yen

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