1 ECON 240C Lecture 8
2
Part I. Economic Forecast Project
• Santa Barbara County Seminar– April 17, 2003
• URL: http://www.ucsb-efp.com
10
One Period Ahead Forecast
• Note the standard error of the regression is 0.2237
• Note: the standard error of the forecast is 0.2248
• Diebold refers to the forecast error– without parameter uncertainty, which will just
be the standard error of the regression– or with parameter uncertainty, which accounts
for the fact that the estimated intercept and slope are uncertain as well
11
Parameter Uncertainty
• Trend model: y(t) = a + b*t + e(t)
• Fitted model: tbay *ˆˆˆ tbaty *ˆˆ)(ˆ
18
Variance of the Forecast Error
)1Re()1(*ˆ)1(*ˆˆ*2]ˆ[
)1(ˆ)1(ˆ[2
tVAtbVARtbaCOVaaVAR
tyEtyVAR t
0.000501 +2*(-0.00000189)*398 + 9.52x10-9*(398)2 +(0.223686)2
0.000501 - 0.00150 + 0.001508 + 0.0500354 0.505444SEF = (0.505444)1/2 = 0.22482
22
Is the Mean Fractional Rate of Growth Different from Zero?
• Econ 240A, Ch.12.2
• where the null hypothesis is that = 0.
• (0.008625-0)/(0.045661/3971/2)
• 0.008625/0.002292 = 3.76 t-statistic, so 0.008625 is significantly different from zero
)//()( nsx
23
Model for lnsp500(t)
• Lnsp500(t) = a +b*t +resid(t), where resid(t) is close to a random walk, so the model is:
• lnsp500(t) a +b*t + RW(t), and taking expontial
• sp500(t) = ea + b*t + RW(t) = ea + b*t eRW(t)
24
Part III. Autoregressive Representation of a Moving Average Process
• MAONE(t) = WN(t) + a*WN(t-1)
• MAONE(t) = WN(t) +a*Z*WN(t)
• MAONE(t) = [1 +a*Z] WN(t)
• MAONE(t)/[1 - (-aZ)] = WN(t)
• [1 + (-aZ) + (-aZ)2 + …]MAONE(t) = WN(t)
• MAONE(t) -a*MAONE(t-1) + a2 MAONE(t-2) + .. =WN(t)
27
Part IV. Significance of Autocorrelations
•
x, x (u) ~ N(0, 1/T) , where T is # of observations x, x (u) ~ N(0, 1/T) , where T is # of observations
29
Box-Pierce Statistic
xx,)(ˆ
)/1/()0)(ˆ(
,
,
uT
Tu
xx
xx
Is normalized, 1.e. is N(0,1)
The square of N(0,1) variables is distributed Chi-square
)(ˆ ,2 uT xx
30
Box-Pierce StatisticThe sum of the squares of independent N(0, 1) variables is Chi-square, and if the autocorrelations are close to zero they will be independent, so under the null hypothesis that the autocorrelations are zero, we have a Chi-square statistic:
)(ˆ1
,2 uT
K
u
xx
that has K-p-q degrees of freedom where K is the number of lags in the sum, and p+q are the number of parameters estimated.
31
Application to Lab Four: the Fractional Change in the Federal Funds Rate
• Dlnffr = lnffr-lnffr(-1)
• Does taking the logarithm and then differencing help model this rate??
35
How would you model dlnff(t) ?
• Notation (p,d,q) for ARIMA models where d stands for the number of times first differenced, p is the order of the autoregressive part, and q is the order of the moving average part.
38
Calculating the Box-Pierce Stat
Lag ACF ACF square SUM Sum*5841 0.013 0.000169 0.000169 0.0986962 -0.015 0.000225 0.000394 0.2300963 -0.026 0.000676 0.00107 0.624884 -0.004 0.000016 0.001086 0.6342245 -0.029 0.000841 0.001927 1.125368
40
Q-Stat at Lag 5
• (T+2)/(T-5) * Box-Pierce = Ljung-Box
• (586/581)*1.25368 = 1.135 compared to 1.132(EVIEWS)