A first important step before composing the VAR(p) models is to study the individual dynamics of each time series to be used in the system. This section investigates univariate model specifications of inflation and nominal interest rates, including nonstationarity tests in the presence of structural breaks.
The data are analyzed in annualized quarterly frequency. The three-month Treasury bill rate is used as the nominal interest rate, whereas the log of the first difference of the seasonally adjusted Consumer Price Index (CPI) is used as a measure of inflation.2 The analysis is carried out for two sample periods: from the second quarter of 1960 to the third quarter of 1995 (142 observations), and from the second quarter of 1960 to the second quarter of 2004 (177 observations).3
Each series, inflation and nominal interest rates, is first tested in order to identify potential time intervals containing structural breaks. The recursive residuals test, CUSUM test, CUSUMQ test, and the recursive coefficients test are applied to an AR(1) model of the
2 The data are obtained from the St. Louis F.R.E.D. For CPI (CPIAUCSL) and the 3-month T-Bill rates (TB3MS), monthly data are converted to quarterly data.
3 The first quarter of 1960 is lost due to the construct of inflation.
The Nonparametric Time-Detrended Fisher Effect 63 series.4 Once an interval containing a potential structural break is identified by at least one of the previously mentioned tests, the log likelihood ratio of the Chow breakpoint test and the Chow forecast test is then applied to specific times within the indicated periods. Based upon agreement of the two Chow tests for both sample periods, a structural break is found in 1981:Q3 for inflation and in 1980:Q2 for nominal interest rates.
The optimal univariate model for each time series is chosen taking into consideration the parsimony of lag length using the Akaike Information Criteria (AIC) and the Schwartz’s Bayesian Criteria (SBC) and the statistical significance of each regressor as well as the whiteness of the residuals.
The Augmented Dickey-Fuller (ADF) test is used to test for stationarity of inflation and nominal interest rates. The test indicates that inflation and nominal interest rates do not have unit roots for the first sample period (1960:Q2-1995:Q3); it fails to reject the hypothesis of nonstationarity in these series for the second sample period (1960:Q2 and 2004:Q2). A summary of the results is reported in Tables 1A and 1B.
The conflicting results in the unit root tests across samples point to a possible model misspecification, which could lead the ADF test to erroneously fail to reject the null of nonstationarity. Hence, the dynamics of the series are further investigated by including an appropriate deterministic trend. Based on the findings of breakpoints in these series, Perron’s test (1989) for nonstationarity against the alternative of a deterministic trend in the presence of sudden structural changes is used in the next step.
Table 1A. INFLATION (πt)—Results of the ADF Test
Sample
Period Model
Estimated Coefficient
of φ
Estimated t-statistic
of φˆ
ADF Critical
Value
Result Unit Root
Process (Yes or No) 1960:Q2 -
1995:Q3
Intercept Only (3 lags Δπt) Removing Δπt-1
-0.137 -3.2733 -2.882 No
1960:Q2 - 2004:Q2
Intercept Only
(3 lags Δπt) -0.111 -2.758 -2.878 Yes
1960:Q2 - 1979:Q1
Trend & Intercept;
Eliminating Δπt-1 and Δπt-2, (4 lags Δπt)
-0.338 -4.117 -3.473 No
1984:Q4 - 1995:Q3
Intercept Only
(0 lags Δπt) -0.596 -4.167 -2.929 No
1984:Q4 - 2004:Q2
Intercept Only
(0 lags Δπt) -0.538 -5.277 -2.898 No
4 All hypotheses are tested at the 5% significance level.
Heather L.R. Tierney 64
Table 1B. NOMINAL INTEREST RATES (it)—Results of the ADF Test
Sample
Period Model
Estimated Coefficient of
φ
Estimated t-statistic
of φˆ
ADF Critical
Value
Result Unit Root Process
(Yes or No)
1960:Q2 - 1995:Q3
Intercept Only (5 lags Δit) Removing Δit-4
-0.083 -3.241 -2.882 No
1960:Q2 - 2004:Q2
Intercept Only (5 lags Δit) Removing Δit-4
-0.111 -2.815 -2.878 Yes
1960:Q2 - 1979:Q1
Trend &
Intercept (3 lags Δit) Removing Δit-2
-0.237 -3.952 -3.471 No
1984:Q4 - 1995:Q3
Trend &
Intercept (6 lags Δit) Removing Δit-2, Δit-3, Δit-4, Δit-5
-0.227 -4.746 -3.514 No
1984:Q4 - 2004:Q2
Trend &
Intercept (6 lags Δit) Removing Δit-2, Δit-3, Δit-4, Δit-5
-0.159 -4.691 -3.467 No
The inclusion of a time trend is just as important in Perron’s test since otherwise it could also mistakenly lead to failing to reject the null of nonstationarity. The test is estimated under the alternate hypothesis of trend stationarity in the residuals of the detrended series. For this reason, three specific types of deterministic time trends are considered as alternate hypotheses: Model A – taking into account a break in the mean (intercept); Model B – taking into account a break in the drift (slope); and Model C – taking into account a break in the mean and in the drift. Model C, which encompasses Models A and B, is:
0 2 2 3 a
t L T t
x =a +a t+μ D +μ D +x (1) where xt is either nominal interest rates or inflation, t refers to the time trend, andxta are the residuals of the detrended series. DL is a dummy variable that takes a value of 0 for t < TB and a value of 1 for t ≥ TB, where TB is the time of the structural break. DT is the dummy variable, which is DL multiplied by the time trend that takes on a value of 0 for t < TB, and a value of the time trend, t, for any t ≥ TB.5
The results of Perron’s test are reported in Tables 2A and 2B. The test selects Model C as the best specification for both series and for both sample periods. This result is illustrated in
5 A more complete report of the results for Perron’s test can be found in Tables 2A and 2B.
The Nonparametric Time-Detrended Fisher Effect 65 Graphs 1A and 1B. Since the residuals of the detrended regressions of inflation and nominal interest rates, xta, are found to be trend stationary, these detrended residuals – which take into account the structural breaks found in the individual series – will be used in the VAR in the next section.
Graph 1A. Inflation; 1960:Q2 to 2004:Q2.
Graph 1B. Nominal Interest Rates; 1960:Q2 to 2004:Q2.
Table 2A. DETRENDED INFLATION (πta) Results of the Perron Test for Structural Change
Sample
Period Time Series Model
Break Fraction1
λ
Estimated Coefficient
of a1
Estimated t-statistic of
ˆ1
a
Perron Critical Value
Result:
Unit Root (Yes or No)
1960:Q2 to 1995:Q3
a
πt
Model A;
Eliminating
1 a
πt−
Δ
(3 lags Δπta)
0.60 0.715 -4.775 -3.76 No
1960:Q2 to 1995:Q3
a
πt
Model B;
Eliminating
1 a
πt−
Δ , &
2 a
πt−
Δ
(3 lags Δπta)
0.60 0.572 -6.381 -3.95 No
1960:Q2 to 1995:Q3
a
πt
Model C;
Eliminating
1 a
πt−
Δ , & Δπta−2
(3 lags Δπta)
0.60 0.550 -6.622 -4.24 No
1960:Q2 to 2004:Q2
a
πt Model A (3 lags a
πt
Δ ) 0.50 0.853 -3.015 -3.76 Yes
1 The break fraction, λ, is rounded off to the tenths since only these critical values are provided in Perron (1989). For the sample period of 1960:Q2 to 1995:Q3 for inflation, λ = 85/142 =0.598, and for nominal interest rates, λ = 80/142 =0.563. For the sample period of 1960:Q2 to 1995:Q3 for inflation, λ = 85/177 =0.480, and for nominal interest rates, λ = 80/177 =0.452.
Table 2A. Continued
Sample Period
Time
Series Model
Break Fraction2
λ
Estimated Coefficient
of a1
Estimated t-statistic of
ˆ1
a
Perron Critical Value
Result:
Unit Root (Yes or No)
1960:Q2 to 2004:Q2
a
πt
Model B;
Eliminating
1 a
πt−
Δ , & Δπta−2
(3 lags Δπta)
0.50 0.609 -6.755 -3.96 No
1960:Q2 to 2004:Q2
a
πt
Model C Eliminating
1 a
πt−
Δ ,Δπta−2
(3 lags Δπta)
0.50 0.529 -7.525 -4.24 No
Table 2B. Detrended Nominal Interest Rates (ita) Results of the Perron Test for Structural Change
Sample Period
Time
Series Model
Break Fraction
λ
Estimated Coefficient
of a1
Estimated t-statistic of aˆ1
Perron Critical
Value
Result:
Unit Root (Yes or No) 1960:Q2
to 1995:Q3
a
it
Model A;
Eliminating
4 a
it−
Δ
(5 lags Δita)
0.50 0.908 -3.090 -3.76 Yes
2 The break fraction, λ, is rounded off to the tenths since only these critical values are provided in Perron (1989). For the sample period of 1960:Q2 to 1995:Q3 for inflation, λ = 85/142 =0.598, and for nominal interest rates, λ = 80/142 =0.563. For the sample period of 1960:Q2 to 1995:Q3 for inflation, λ = 85/177 =0.480, and for nominal interest rates, λ = 80/177 =0.452.
Table 2B. Continued
Sample Period
Time
Series Model
Break Fraction
λ
Estimated Coefficient
of a1
Estimated t-statistic of aˆ1
Perron Critical Value
Result:
Unit Root (Yes or No)
1960:Q2 to 1995:Q3
a
it
Model B;
Eliminating
4 a
it−
Δ
(5 lags Δita)
0.50 0.906 -3.487 -3.95 Yes
1960:Q2 to 1995:Q3
a
it
Model C;
Eliminating
4 a
it−
Δ (5 lags Δita)
0.50 0.691 -5.005 -4.24 No
1960:Q2 to 2004:Q2
a
it
Model A;
Eliminating
4 a
it−
Δ
5 lags Δita)
0.50 0.917 -2.987 -3.76 Yes
1960:Q2 to 2004:Q2
a
it
Model B;
Eliminating
4 a
it−
Δ , Δita−5, & Δita−6
(7 lags Δita)
0.50 0.953 -2.139 -3.96 Yes
1960:Q2 to 2004:Q2
a
it
Model C Eliminating
4 a
it−
Δ (5 lags Δita)
0.50 0.775 -5.237 -4.24 No
The Nonparametric Time-Detrended Fisher Effect 69