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Some analysis of the long-run time series properties of consumption and income in the U.K.
Economics Letters North-Holland
39(1992)
173-17X
173
Some analysis of the long-run time series properties of consumption and income in the U.K. D.A. Peel Uttr~wsii): College of Wales, Aberysiuyth, Received 4ccepted
UK
13 November 1991 17 January 1992
The purpose in this letter is to examine the long-run time series behaviour of consumption and income in the UK. Both series are non-stationary and exhibit non-linearity in conditional mean. Evidence is provided of cointegration and non-linear error correction.
1.
Introduction
of real consumption The purpose in this note is to examine the long-run time series properties and income in the U.K. employing annual data spanning the period 1830 to 1990. The motivation for our analysis to twofold. Firstly, there are many theoretical models which are suggestive that either real income or consumption could have non-linear time series representations [see, e.g. Day (1983)]. Consequently it will be of interest to examine whether the long-run data for the U.K. exhibits non-linear properties. Second, in much empirical work relating consumption and income it is typically, implicitly, assumed that the variables have linear time series representations. For example, ‘error correction mechanisms’ linking consumption and income are given a linear form [see, e.g. Molana (199111. However, if consumption and income are both integrated processes, but have non-linear time series representations, then linear error correction mechanisms - which in general imply linear univariate time series representations - will not fully capture the dynamic process relating consumption and income. Recently a number of authors have begun to report non-linear error correction mechanisms [see, e.g. Granger and Lee (1990>, Escribano and Pfann (lY90)] though not as yet for the consumption income process.
2. The data and its time series properties Our data is Real Consumer Expenditure and Gross National Product as Market Prices from Mitchell (1990) and from Economic Trends giving 161 observations. We estimate models over the Correspondence to: D.A. Peel, Department Llandinam Building, Penglais, Aberystwyth, 01651765/92/$05.00
0 1992 - Elsevier
of Economics and Agricultural Dyfed SY23 3DB, UK.
Science
Publishers
Economics,
B.V. All rights reserved
The University
College
of Wales,
D.A. Peel / Analysis of long-run time series properties
174 Table 1 Summary
statistics
for series and residuals,
1836-1985.
a
Variable
Variance
Skewness
Kurtosis
JB
LB,
LB,
A(l)
A(4)
DY DC DCR DCRA DYG DCRAG DYB DYBG DCRB
11.95 7.57 6.62 6.08 1.00 1.00 10.74 1.00 5.20
- 0.582 - 0.654 0.245 0.11 -0.107 -0.03 - 0.03 -0.107 0.11
5.00 8.19 6.34 5.64 4.22 4.73 4.15 4.22 6.16
33.67 185.21 11.27 44.02 10.01 18.80 8.42 10.01 62.91
1.52 5.85 2.09 0.01 0.08 0.55 0.02 0.02 0.40
6.75 8.90 12.83 0.87 3.59 2.20 6.29 5.81 3.15
31.04 13.45 13.76 13.21 0.53 0.38 15.51 0.47 0.09
37.15 21.4 21.48 14.83 5.50 2.67 21.59 5.09 0.95
i1 DY, DC, DCR are respectively the percentage rates of change in logarithms of real income, consumption and consumption residuals. G denotes errors for joint AR and ARCH process. B - bilinear process. Consequently DYB are the residuals from the estimated bilinear process for real income. LB, and LB, are the Ljung-Box Q statistic with 1 and 4 lagged autocorrelations. A(1) and A(4) are Engle’s (1982) ARCH test with 1 and 4 squared residuals. JB is the Jarque-Bera normality test statisttc which is distributed as x’.
period 1830-1985 and retain five observations to evaluate the forecasting properties of the various estimated models. In table 1 we report some basic properties of the series. These are the Ljung-Box Q statistic for the autocorrelations of the log of first differences, a test for ARCH, given in Engle (1982), the Jarque-Bera test statistic for departure from normality, as well as the skewness and kurtosis coefficients. We do not report the levels autocorrelations, which all start high, the lowest first log sample autocorrelation is 0.97, and decline quite slowly, supporting the presumption of nonstationarity. This was confirmed by more formal empirical tests, namely Phillips-Perron test statistics computed following the procedures outlined in Perron (1988). These statistics are particularly appropriate if a non-linear data generating mechanism is a priori feasible, since the test statistics are robust in the presence of a heterogeneous non-independent error process in the equations employed to derive the test statistics. In no case is the null hypothesis not rejected (this was also the case for a number of subsets of data we analysed). All the log differenced series display excess kurtosis, a feature which is to be expected in the context of significant ARCH effects - which are present - but could be indicative of other types of non-linearity. The non-normality of changes in consumption is particularly marked. We experimented with two dummy variables to allow for any impacts of World War One and World War Two on the processes. For changes in consumption these two dummies were highly significant, but not for output. We report results for the residuals obtained from the regression of consumption on the two dummies. These are denoted DCR. We observe from table 1 that these residuals still exhibit significant non-normality. Additionally, the DCR residuals exhibit significant serial correlation on the basis of the Ljung-Box statistic. An autoregressive process of order two generates white noise residuals (denoted DC&l) though they still exhibit significant non-normality. Weiss (1986) demonstrates how a linear process with ARCH errors can mimic a bilinear process. The bilinear process is discussed in, for example, Subba Rao and Gaber (1984). It can be demonstrated that a bilinear process is a reasonably general non-linear model in that it is an arbitrarily close second-order approximation to any underlying non-linear process. The bilinear process with no moving average terms is given by P Y, +
C aiYt-j=a
i=l
P
Q
+ C C bijY,-ic,-, i=l
j=l
+ E,?
(1)
D.A. Peel /Analysis Table 2 Bilinear models. Series
of long-run time series properties
175
”
Bilinear
Constant
Parameters
of y,,
0.119
P
P
Q
DY DC‘R
1 2
2 1
1 5
2.06 2.312
-0.133 - 0.206
VfRCR
1
5
2
0.0046
- 0.822
Parameters
of y,,
- 0.044 - 0.067 0.061 - 2.93 - 8.69
- 0.027 -0.01 - 0.095 - 3.40 -2.14
I?(,,
- 0.05
- 0.049
- 2.55 - 1.46
- 1.59 - 7.22
7.72
12.5
’ .MRCR are the residuals
obtained from the regression of the cointegration residuals from (2) regressed on the two war dummies. The coefficients in the bilinear process are interpreted with reference to eq. (1). For instance, DY has p = 1 (one autoregressive term), P = 5, Q = 1. Consequently the process has the form DY = Y, = 2.06 + 0.133 Y_, - 0.044 Y,_ ,. F,_, -0.027 Yl_Z~,_, +E!.
where Ed is a sequence of i.i.d. random variables and p, P, Q, a,,a and bi are real constants. The bilinear models are estimated employing a computer program developed by Subba Rao. ’ We set the maximum values of p, P and Q at 5 and allowed the programme to select the most parsimonious mode1 on the basis of the AIK criteria. Standard errors for the coefficients are not reported in the program. In table 1 we also report the summary statistics for the estimated processes. The bilinear model generates significant reductions in residual variance relative to the most parsimonious linear process. For real output this is some 10% and for consumption 16%. The most parsimonious bilinear models are reported in table 2. The residuals from the bilinear mode1 still exhibit excess kurtosis and significant ARCH effects. Accordingly we experimented with ‘second generation models’ [the term used by Tong (1990)] which are mixed models, e.g. bilinear with ARCH errors. Significant joint models were estimated. The residuals from the models appear to still depart from normality on the basis of the Jarque-Bera test. These empirical results are thus suggestive that over the sample period 1831-1985 consumption and income exhibited significant conditional mean non-linearity, with some evidence of first-order ARCH effects. We next examined whether (the logs of) consumption and income were cointegrated over our sample period. We employed the estimator suggested by Phillips and Hansen (1990). This is an asymptotically median-unbiased estimator and appears to have good operating characteristics in small samples. We obtained:
1830-198.5 LC = 0.598 + 0.91 (0.103)
LY, R2 = 0.9918,
(2)
(0.0091)
where LC, LY are respectively standard errors in parentheses.
the logarithm
of real consumption
and real income;
asymptotic
’ We also experimented with threshold autoregressive models [see, e.g. Tong (199O)I. These models are particularly suitable for data exhibiting limit cycles. However, for our data the threshold models had higher in-sample variance of residuals than the bilinear models, particularly for consumption.
DY
DY
DY
DY
DY
DY
DY
DC
DC
DC
1920-1985
1832-1985
1832-1985
1832-1985
1920-1985
1832-1985
1920-1985
1832-1985
1832-1985
1920-1985
2.067 (0.41) 1.616 (0.58) 1.89 (0.43) 2.41 (0.36) 2.58 (0.35) 2.56 (0.60) 2.92 (0.38) 3.26 (0.67) 2.31 (0.33) 2.56 (0.34) 2.74 (0.54)
Constant
a
- 6.23 (2.49) - 5.23 (2.47)
0.93 (1.30)
0.50 (1.62) 0.64 (1.42)
WWI
4.81 (2.08) 4.05 (2.86) - 5.54 (2.34) - 5.32 (2.30) - 5.37 (2.91)
2.65 (2.10) 3.44 (2.09)
wW2
18.92 (4.46) 14.18 (4.28) 23.30 (5.95) 24.15 (6.08) 18.50 (4.40) 13.54 (4.38) 35.11 (6.87) 33.96 (10.88) - 11.84 (6.19) - 14.35 (6.13) - 14.49 (7.60) - 25.87 (8.57) - 31.99 (12.03)
(DRES)
(-1)
RES C-1)
DY
0.204 (0.124) 0.436 (0.14) 0.187 (0.131) 0.26 (0.11) 0.28 (0.10) 0.41 (0.10) 0.22 (0.10) 0.32 (0.10)
C-1) - 0.337 (0.133) - 0.281 (0.16) - 0.296 (0.158) -0.31 (0.13) - 0.36 (0.12) - 0.38 (0.13) -0.31 (0.12) - 0.36 (0.14) 0.09 (0.13) 0.19 (0.10) 0.28 (0.16) - 0.21 (0.10) - 0.21 (0.09) - 0.21 (0.11)
t_c2)
- 0.04 (0.01) - 0.04 (0.01) - 0.04 (0.01) - 0.05 (0.01) - 0.06 (0.01)
DY2 (-1)
0.24
0.43
0.25
0.37
0.20
0.22
0.14
0.29
0.14
R2
-0.04 0.30 (0.01) - 0.037 0.31 (0.018)
C-1)
DC2
5.74
1.78
1.17
6.78
2.73
5.17
2.58
1.92
3.40
5.43
3.48
LB,
7.52
15.97
38.23
0.17
9.79
0.01
7.18
8.68
15.62
1.20
10.52
A(1)
3.14
3.41
3.97
3.83
3.45
4.29
3.66
3.36
3.67
4.02
3.89
Kurtosis
0.45
1.13
6.14
3.70
1.35
4.59
2.92
0.88
6.58
3.06
6.96
JB
errors: Heteroskedastic consistent via White’s (1980) method in parentheses. DY is the difference in the logarithm of real output x 100. DC is the in the logarithm of real consumption X 100. RES is the residual from the cointegrating regressions. DRES is a dummy variable times RES where the defined as 1, when RES > 0, and 0 when RES < 0. DC2 and DY2 are respectively the square of changes in the log of consumption and income. wW1 and 1, 0 dummies which allow for the First and Second World Wars.
DY
1832-1985
’ Standard difference dummy is wW2 are
Dep. var.
mechanisms.
Period
Table 3 Error correction
177
D.A. Peel / Analysis of long-run time seriesproperties
The Phillips/Perron
Z(&)
test statistics
= -33.94
for the residual
from eq. (2) gave [Perron
Same to third decimal included
Z( t&) = -4.199
point
(1988) notation]:
if time trend
as regressor
We therefore accept the hypothesis of cointegration. In table 2 we report the bilinear process estimated for the residuals from the cointegrating regression having first dummied out the effects of the war. They are described by a significant bilinear process, which relative to the most parsimonious linear process [an AR(l)], reduces residual variance by a substantial 45%. Given the significant non-linearity, we examined a variety of non-linear error-correction mechanisms. In particular we examined the significance of squares of real income and consumption rates of change, as well as allowing for non-symmetric speeds of adjustment. This was achieved by allowing the response of changes in income or consumption to the lagged residual from the cointegrating regression to have differential response to positive or negative residuals. A selection of results are reported in table 3. The key feature of the results, apart from the significance of the lagged cointegrating residual which provides additional support for cointegration, is the significance of non-linear terms in the error correction mechanisms. In particular, lagged changes in the square of real output change are significant in the error correction mechanism for output and lagged changes in the square of real consumption in the error correction mechanism for consumption. In addition, there is some evidence of significant asymmetric adjustment in the error correction mechanism for output. Although the non-linear time series models and the non-linear error corrections provide a significant improvement in terms of in-sample residual variance relative to the linear models, they may not, of course, provide better predictions out-of-sample. There is an empirical finding reported in other analyses which support non-linear models in-sample [see, e.g., Diebold and Nason (199011. To provide some (limited) evidence on this issue we examined the forecasting performance of the bilinear models and non-linear error correction models (fitted over the period 1832-1985) relative to the linear models on a hold-out sample of five observations. The mean square errors of the various forecasts are reported in table 4. We observe from table 4 that although the bilinear models have lower mean square errors than any competing models, the non-linear error corrections produce worse forecasts than the linear error corrections, which are themselves inferior to the linear autoregressive models.
Table 4 Mean square
errors,
Series
No change
Bilinear
Linear autoregressive
Linear error correction
Nonlinear correction
Non-linear assymetric
DY DCR DC
10.82 9.86 24.34
2.85 8.95 _
3.24 10.49 11.85
3.55 _ 16.91
4.00 _ 21.68
3.67 _ _
1986-1990
178
D.A. Peel / Analysis of long-run time series properties
3. Conclusions
The purpose in this note has been to determine whether consumption and income in the U.K. over the period 1830-1985 exhibited significant non-linearity. Bilinear models were found to produce significant reductions in in-sample residual variance over linear alternatives, though the residuals still exhibited significant departure from normality. Consumption and income appeared to be cointegrated series and given their non-linear time series representations we experimented with non-linear error corrections. We found evidence of significant non-linearity in-sample but the out-of-sample forecasts were significantly worse than the bilinear or linear time series representations. Clearly in future work it will be of interest to examine alternative non-linear error correction mechanisms.
References Day, R., 1983, The emergence of chaos from classical economic growth, Quarterly Journal of Economics. Diebold, F.X. and J. Nason, 1990, Nonparametric exchange rate prediction, Journal of International Economics. Engle, R.F., 1982, Autoregressive conditional heteroskedasticity with estimates of the variance of United Kingdom inflation, Econometrica. Engle, R.F. and C.W.J. Granger, 1989, Dynamic model specification with equilibrium constraints, cointegration and error correction, Econometrica. Escribano, A. and G. Pfann, 1990, Nonlinear error correction, asymmetric adjustment and cointegration, Mimeo. (University of Limburg, Maastricht, The Netherlands). Granger, C.W.J. and T.H. Lee, 1990, Investigation of production, sales and inventory relationships using multicointegration and nonsymmetric error correction models, Journal of Applied Econometrics. Mitchell, B.R., 1990, British historical statistics (Cambridge University Press, Cambridge). Molana, H., 1991, The time series consumption functions: error correction, random walk and the steady state, The Economic Journal. Perron, P., 1988, Trends and random walks in macroeconomic time series, Journal of Economic Dynamics and Control. Phillips, P.C.B. and B. Hansen, 1990, Statistical inferences in instrumental variables regressions with I(1) processes, Review of Economic Studies. Subba Rao, T. and M.M. Fabr, 1984, An introduction to bispectral analysis and bilinear time series models, Lecture Notes in Statistics, Vol. 24 (Springer, New York). Tong, H., 1990, Non-linear times series: A dynamical system approach (Oxford Science Publications, Clarendon Press, Oxford). Weiss, A.A., 1986, ARCH and bilinear time series models: Comparison and combination, Journal of Business and Economic Statistics. White, H., 1980, A heteroskedasticity-consistent covariance matrix and direct test for heteroskedasticity, Econometrica.
39(1992)
173-17X
173
Some analysis of the long-run time series properties of consumption and income in the U.K. D.A. Peel Uttr~wsii): College of Wales, Aberysiuyth, Received 4ccepted
UK
13 November 1991 17 January 1992
The purpose in this letter is to examine the long-run time series behaviour of consumption and income in the UK. Both series are non-stationary and exhibit non-linearity in conditional mean. Evidence is provided of cointegration and non-linear error correction.
1.
Introduction
of real consumption The purpose in this note is to examine the long-run time series properties and income in the U.K. employing annual data spanning the period 1830 to 1990. The motivation for our analysis to twofold. Firstly, there are many theoretical models which are suggestive that either real income or consumption could have non-linear time series representations [see, e.g. Day (1983)]. Consequently it will be of interest to examine whether the long-run data for the U.K. exhibits non-linear properties. Second, in much empirical work relating consumption and income it is typically, implicitly, assumed that the variables have linear time series representations. For example, ‘error correction mechanisms’ linking consumption and income are given a linear form [see, e.g. Molana (199111. However, if consumption and income are both integrated processes, but have non-linear time series representations, then linear error correction mechanisms - which in general imply linear univariate time series representations - will not fully capture the dynamic process relating consumption and income. Recently a number of authors have begun to report non-linear error correction mechanisms [see, e.g. Granger and Lee (1990>, Escribano and Pfann (lY90)] though not as yet for the consumption income process.
2. The data and its time series properties Our data is Real Consumer Expenditure and Gross National Product as Market Prices from Mitchell (1990) and from Economic Trends giving 161 observations. We estimate models over the Correspondence to: D.A. Peel, Department Llandinam Building, Penglais, Aberystwyth, 01651765/92/$05.00
0 1992 - Elsevier
of Economics and Agricultural Dyfed SY23 3DB, UK.
Science
Publishers
Economics,
B.V. All rights reserved
The University
College
of Wales,
D.A. Peel / Analysis of long-run time series properties
174 Table 1 Summary
statistics
for series and residuals,
1836-1985.
a
Variable
Variance
Skewness
Kurtosis
JB
LB,
LB,
A(l)
A(4)
DY DC DCR DCRA DYG DCRAG DYB DYBG DCRB
11.95 7.57 6.62 6.08 1.00 1.00 10.74 1.00 5.20
- 0.582 - 0.654 0.245 0.11 -0.107 -0.03 - 0.03 -0.107 0.11
5.00 8.19 6.34 5.64 4.22 4.73 4.15 4.22 6.16
33.67 185.21 11.27 44.02 10.01 18.80 8.42 10.01 62.91
1.52 5.85 2.09 0.01 0.08 0.55 0.02 0.02 0.40
6.75 8.90 12.83 0.87 3.59 2.20 6.29 5.81 3.15
31.04 13.45 13.76 13.21 0.53 0.38 15.51 0.47 0.09
37.15 21.4 21.48 14.83 5.50 2.67 21.59 5.09 0.95
i1 DY, DC, DCR are respectively the percentage rates of change in logarithms of real income, consumption and consumption residuals. G denotes errors for joint AR and ARCH process. B - bilinear process. Consequently DYB are the residuals from the estimated bilinear process for real income. LB, and LB, are the Ljung-Box Q statistic with 1 and 4 lagged autocorrelations. A(1) and A(4) are Engle’s (1982) ARCH test with 1 and 4 squared residuals. JB is the Jarque-Bera normality test statisttc which is distributed as x’.
period 1830-1985 and retain five observations to evaluate the forecasting properties of the various estimated models. In table 1 we report some basic properties of the series. These are the Ljung-Box Q statistic for the autocorrelations of the log of first differences, a test for ARCH, given in Engle (1982), the Jarque-Bera test statistic for departure from normality, as well as the skewness and kurtosis coefficients. We do not report the levels autocorrelations, which all start high, the lowest first log sample autocorrelation is 0.97, and decline quite slowly, supporting the presumption of nonstationarity. This was confirmed by more formal empirical tests, namely Phillips-Perron test statistics computed following the procedures outlined in Perron (1988). These statistics are particularly appropriate if a non-linear data generating mechanism is a priori feasible, since the test statistics are robust in the presence of a heterogeneous non-independent error process in the equations employed to derive the test statistics. In no case is the null hypothesis not rejected (this was also the case for a number of subsets of data we analysed). All the log differenced series display excess kurtosis, a feature which is to be expected in the context of significant ARCH effects - which are present - but could be indicative of other types of non-linearity. The non-normality of changes in consumption is particularly marked. We experimented with two dummy variables to allow for any impacts of World War One and World War Two on the processes. For changes in consumption these two dummies were highly significant, but not for output. We report results for the residuals obtained from the regression of consumption on the two dummies. These are denoted DCR. We observe from table 1 that these residuals still exhibit significant non-normality. Additionally, the DCR residuals exhibit significant serial correlation on the basis of the Ljung-Box statistic. An autoregressive process of order two generates white noise residuals (denoted DC&l) though they still exhibit significant non-normality. Weiss (1986) demonstrates how a linear process with ARCH errors can mimic a bilinear process. The bilinear process is discussed in, for example, Subba Rao and Gaber (1984). It can be demonstrated that a bilinear process is a reasonably general non-linear model in that it is an arbitrarily close second-order approximation to any underlying non-linear process. The bilinear process with no moving average terms is given by P Y, +
C aiYt-j=a
i=l
P
Q
+ C C bijY,-ic,-, i=l
j=l
+ E,?
(1)
D.A. Peel /Analysis Table 2 Bilinear models. Series
of long-run time series properties
175
”
Bilinear
Constant
Parameters
of y,,
0.119
P
P
Q
DY DC‘R
1 2
2 1
1 5
2.06 2.312
-0.133 - 0.206
VfRCR
1
5
2
0.0046
- 0.822
Parameters
of y,,
- 0.044 - 0.067 0.061 - 2.93 - 8.69
- 0.027 -0.01 - 0.095 - 3.40 -2.14
I?(,,
- 0.05
- 0.049
- 2.55 - 1.46
- 1.59 - 7.22
7.72
12.5
’ .MRCR are the residuals
obtained from the regression of the cointegration residuals from (2) regressed on the two war dummies. The coefficients in the bilinear process are interpreted with reference to eq. (1). For instance, DY has p = 1 (one autoregressive term), P = 5, Q = 1. Consequently the process has the form DY = Y, = 2.06 + 0.133 Y_, - 0.044 Y,_ ,. F,_, -0.027 Yl_Z~,_, +E!.
where Ed is a sequence of i.i.d. random variables and p, P, Q, a,,a and bi are real constants. The bilinear models are estimated employing a computer program developed by Subba Rao. ’ We set the maximum values of p, P and Q at 5 and allowed the programme to select the most parsimonious mode1 on the basis of the AIK criteria. Standard errors for the coefficients are not reported in the program. In table 1 we also report the summary statistics for the estimated processes. The bilinear model generates significant reductions in residual variance relative to the most parsimonious linear process. For real output this is some 10% and for consumption 16%. The most parsimonious bilinear models are reported in table 2. The residuals from the bilinear mode1 still exhibit excess kurtosis and significant ARCH effects. Accordingly we experimented with ‘second generation models’ [the term used by Tong (1990)] which are mixed models, e.g. bilinear with ARCH errors. Significant joint models were estimated. The residuals from the models appear to still depart from normality on the basis of the Jarque-Bera test. These empirical results are thus suggestive that over the sample period 1831-1985 consumption and income exhibited significant conditional mean non-linearity, with some evidence of first-order ARCH effects. We next examined whether (the logs of) consumption and income were cointegrated over our sample period. We employed the estimator suggested by Phillips and Hansen (1990). This is an asymptotically median-unbiased estimator and appears to have good operating characteristics in small samples. We obtained:
1830-198.5 LC = 0.598 + 0.91 (0.103)
LY, R2 = 0.9918,
(2)
(0.0091)
where LC, LY are respectively standard errors in parentheses.
the logarithm
of real consumption
and real income;
asymptotic
’ We also experimented with threshold autoregressive models [see, e.g. Tong (199O)I. These models are particularly suitable for data exhibiting limit cycles. However, for our data the threshold models had higher in-sample variance of residuals than the bilinear models, particularly for consumption.
DY
DY
DY
DY
DY
DY
DY
DC
DC
DC
1920-1985
1832-1985
1832-1985
1832-1985
1920-1985
1832-1985
1920-1985
1832-1985
1832-1985
1920-1985
2.067 (0.41) 1.616 (0.58) 1.89 (0.43) 2.41 (0.36) 2.58 (0.35) 2.56 (0.60) 2.92 (0.38) 3.26 (0.67) 2.31 (0.33) 2.56 (0.34) 2.74 (0.54)
Constant
a
- 6.23 (2.49) - 5.23 (2.47)
0.93 (1.30)
0.50 (1.62) 0.64 (1.42)
WWI
4.81 (2.08) 4.05 (2.86) - 5.54 (2.34) - 5.32 (2.30) - 5.37 (2.91)
2.65 (2.10) 3.44 (2.09)
wW2
18.92 (4.46) 14.18 (4.28) 23.30 (5.95) 24.15 (6.08) 18.50 (4.40) 13.54 (4.38) 35.11 (6.87) 33.96 (10.88) - 11.84 (6.19) - 14.35 (6.13) - 14.49 (7.60) - 25.87 (8.57) - 31.99 (12.03)
(DRES)
(-1)
RES C-1)
DY
0.204 (0.124) 0.436 (0.14) 0.187 (0.131) 0.26 (0.11) 0.28 (0.10) 0.41 (0.10) 0.22 (0.10) 0.32 (0.10)
C-1) - 0.337 (0.133) - 0.281 (0.16) - 0.296 (0.158) -0.31 (0.13) - 0.36 (0.12) - 0.38 (0.13) -0.31 (0.12) - 0.36 (0.14) 0.09 (0.13) 0.19 (0.10) 0.28 (0.16) - 0.21 (0.10) - 0.21 (0.09) - 0.21 (0.11)
t_c2)
- 0.04 (0.01) - 0.04 (0.01) - 0.04 (0.01) - 0.05 (0.01) - 0.06 (0.01)
DY2 (-1)
0.24
0.43
0.25
0.37
0.20
0.22
0.14
0.29
0.14
R2
-0.04 0.30 (0.01) - 0.037 0.31 (0.018)
C-1)
DC2
5.74
1.78
1.17
6.78
2.73
5.17
2.58
1.92
3.40
5.43
3.48
LB,
7.52
15.97
38.23
0.17
9.79
0.01
7.18
8.68
15.62
1.20
10.52
A(1)
3.14
3.41
3.97
3.83
3.45
4.29
3.66
3.36
3.67
4.02
3.89
Kurtosis
0.45
1.13
6.14
3.70
1.35
4.59
2.92
0.88
6.58
3.06
6.96
JB
errors: Heteroskedastic consistent via White’s (1980) method in parentheses. DY is the difference in the logarithm of real output x 100. DC is the in the logarithm of real consumption X 100. RES is the residual from the cointegrating regressions. DRES is a dummy variable times RES where the defined as 1, when RES > 0, and 0 when RES < 0. DC2 and DY2 are respectively the square of changes in the log of consumption and income. wW1 and 1, 0 dummies which allow for the First and Second World Wars.
DY
1832-1985
’ Standard difference dummy is wW2 are
Dep. var.
mechanisms.
Period
Table 3 Error correction
177
D.A. Peel / Analysis of long-run time seriesproperties
The Phillips/Perron
Z(&)
test statistics
= -33.94
for the residual
from eq. (2) gave [Perron
Same to third decimal included
Z( t&) = -4.199
point
(1988) notation]:
if time trend
as regressor
We therefore accept the hypothesis of cointegration. In table 2 we report the bilinear process estimated for the residuals from the cointegrating regression having first dummied out the effects of the war. They are described by a significant bilinear process, which relative to the most parsimonious linear process [an AR(l)], reduces residual variance by a substantial 45%. Given the significant non-linearity, we examined a variety of non-linear error-correction mechanisms. In particular we examined the significance of squares of real income and consumption rates of change, as well as allowing for non-symmetric speeds of adjustment. This was achieved by allowing the response of changes in income or consumption to the lagged residual from the cointegrating regression to have differential response to positive or negative residuals. A selection of results are reported in table 3. The key feature of the results, apart from the significance of the lagged cointegrating residual which provides additional support for cointegration, is the significance of non-linear terms in the error correction mechanisms. In particular, lagged changes in the square of real output change are significant in the error correction mechanism for output and lagged changes in the square of real consumption in the error correction mechanism for consumption. In addition, there is some evidence of significant asymmetric adjustment in the error correction mechanism for output. Although the non-linear time series models and the non-linear error corrections provide a significant improvement in terms of in-sample residual variance relative to the linear models, they may not, of course, provide better predictions out-of-sample. There is an empirical finding reported in other analyses which support non-linear models in-sample [see, e.g., Diebold and Nason (199011. To provide some (limited) evidence on this issue we examined the forecasting performance of the bilinear models and non-linear error correction models (fitted over the period 1832-1985) relative to the linear models on a hold-out sample of five observations. The mean square errors of the various forecasts are reported in table 4. We observe from table 4 that although the bilinear models have lower mean square errors than any competing models, the non-linear error corrections produce worse forecasts than the linear error corrections, which are themselves inferior to the linear autoregressive models.
Table 4 Mean square
errors,
Series
No change
Bilinear
Linear autoregressive
Linear error correction
Nonlinear correction
Non-linear assymetric
DY DCR DC
10.82 9.86 24.34
2.85 8.95 _
3.24 10.49 11.85
3.55 _ 16.91
4.00 _ 21.68
3.67 _ _
1986-1990
178
D.A. Peel / Analysis of long-run time series properties
3. Conclusions
The purpose in this note has been to determine whether consumption and income in the U.K. over the period 1830-1985 exhibited significant non-linearity. Bilinear models were found to produce significant reductions in in-sample residual variance over linear alternatives, though the residuals still exhibited significant departure from normality. Consumption and income appeared to be cointegrated series and given their non-linear time series representations we experimented with non-linear error corrections. We found evidence of significant non-linearity in-sample but the out-of-sample forecasts were significantly worse than the bilinear or linear time series representations. Clearly in future work it will be of interest to examine alternative non-linear error correction mechanisms.
References Day, R., 1983, The emergence of chaos from classical economic growth, Quarterly Journal of Economics. Diebold, F.X. and J. Nason, 1990, Nonparametric exchange rate prediction, Journal of International Economics. Engle, R.F., 1982, Autoregressive conditional heteroskedasticity with estimates of the variance of United Kingdom inflation, Econometrica. Engle, R.F. and C.W.J. Granger, 1989, Dynamic model specification with equilibrium constraints, cointegration and error correction, Econometrica. Escribano, A. and G. Pfann, 1990, Nonlinear error correction, asymmetric adjustment and cointegration, Mimeo. (University of Limburg, Maastricht, The Netherlands). Granger, C.W.J. and T.H. Lee, 1990, Investigation of production, sales and inventory relationships using multicointegration and nonsymmetric error correction models, Journal of Applied Econometrics. Mitchell, B.R., 1990, British historical statistics (Cambridge University Press, Cambridge). Molana, H., 1991, The time series consumption functions: error correction, random walk and the steady state, The Economic Journal. Perron, P., 1988, Trends and random walks in macroeconomic time series, Journal of Economic Dynamics and Control. Phillips, P.C.B. and B. Hansen, 1990, Statistical inferences in instrumental variables regressions with I(1) processes, Review of Economic Studies. Subba Rao, T. and M.M. Fabr, 1984, An introduction to bispectral analysis and bilinear time series models, Lecture Notes in Statistics, Vol. 24 (Springer, New York). Tong, H., 1990, Non-linear times series: A dynamical system approach (Oxford Science Publications, Clarendon Press, Oxford). Weiss, A.A., 1986, ARCH and bilinear time series models: Comparison and combination, Journal of Business and Economic Statistics. White, H., 1980, A heteroskedasticity-consistent covariance matrix and direct test for heteroskedasticity, Econometrica.