We expect that these five time series are stationary. The graphs of the series wage and
price inflation, capacity utilization rates and labor productivity growth, dwt ,dpt ,Vtl ,Vtc , dynt ,
confirm our expectation. In additional we carry out DF unit root test for each series.
The test results are shown in table 2.
Variable |
Sample |
Critical value |
Test Statistic |
dw |
1947:02 TO 2000:04 |
-3.41000 |
-3.74323 |
dp |
1947:02 TO 2000:04 |
-3.41000 |
-3.52360 |
Vl |
1948:02 TO 2000:04 |
-1.95000 |
-0.73842 |
Vc |
1948:02 TO 2000:04 |
-3.41000 |
-4.13323 |
ukbp |
1950:01 TO 2000:04 |
-3.41000 |
-7.09932 |
R |
1955:01 TO 2000:04 |
-1.95000 |
-0.94144 |
Table 2: Summary of DF-Test Results
4.2 Estimation of unrestricted VAR
The unit root test confirms our expectation with the exception of Vtl and rt . Although
the test cannot reject the null of unit root, there is no reason to expect the rate of
unemployment and the federal funds rate as being a unit root process. Indeed we expect
that they are constrained in certain limited ranges, say from zero to 0.3. Due to the lower
power of DF test, the test result should only provide hints that the rate of unemployment
and the federal funds rate have a strong autocorrelation, respectively.
Given this stationarity we can construct a VAR model for these 6 variables to mimic
the DGP of these 6 variables.
dwt |
( c 1 ^ |
b1 |
a11k a12k a13k a14k a15k a16k | |||
dpt |
c2 |
b2 |
P |
a21k a22k a23k a24k a25k a26k | ||
Vtl |
— |
c3 |
+ |
b3 |
d 74+-V |
a31k a32k a33k a34k a35k a36k |
Vtc |
c4 |
b4 |
Z √ k=1 |
a41k a42k a43k a44k a45k a46k | ||
rt |
c5 |
b5 |
a51k a52k a53k a54k a55k a56k | |||
ukbpt |
c6 |
b6 |
a61k a62k a63k a64k a65k a66k |
/ |
dwt-k |
∖ |
/ |
∖ | ||
dpt-k |
/ |
e1t |
∖ | |||
Vl Vt-k Vc Vt-k |
+ |
e2t e3t e4t | ||||
∖ |
rt-k |
) |
e5t |
) |
(13)
To determine the lag length of the VAR we apply sequential likelihood tests. We start
with a lag length of 24, at which the residuals can be taken as WN process. The
sequence likelihood ratio test procedure gives a lag length of 9. The test results are
listed as follows.
• H0 : P = 20 v.s. H1 : P =24
Chi-Squared(144)= 150.01095 with Significance Level 0.34880
• H0 : P = 16 v.s. H1 : P =20
Chi-Squared(144)= 148.82953 with Significance Level 0.37424
• H0 : P = 12 v.s. H1 : P =16
Chi-Squared(25)= 120.97749 with Significance Level 0.91874
18