- Email: [email protected]
Inflation in South Africa. A long memory approach
Economics Letters 111 (2011) 207–209
Contents lists available at ScienceDirect
Economics Letters j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e c o l e t
Inflation in South Africa. A long memory approach☆ Luis A. Gil-Alana ⁎ University of Navarra, Faculty of Economics, Pamplona, Spain
a r t i c l e
i n f o
Article history: Received 25 March 2010 Received in revised form 20 January 2011 Accepted 15 February 2011 Available online 21 February 2011
a b s t r a c t This paper deals with the analysis of the inflation rate in South Africa for the time period 1970M1–2008M12. We use long range dependence techniques and the results show that inflation in this country is a covariance stationary process with long range dependence, with an order of integration ranging in the interval (0, 0.5). Policy implications are derived. © 2011 Elsevier B.V. All rights reserved.
JEL classification: C22 Keywords: Inflation South Africa Long range dependence
1. Introduction Long range dependence or long memory is a characteristic of many economic time series including inflation rates. (See, for example, the paper of Backus and Zin, 1993, for the US case; and Hassler, 1993, and Delgado and Robinson, 1994, for the Swiss and Spanish inflation rates respectively). In this paper we use fractional integration to show that the inflation rate in South Africa may also be characterized as an I(d) process with the value of d ranging in the interval (0, 0.5). Thus, inflation in this country may be well described as a covariance stationary process with long range dependence. The structure of the article is as follows: Section 2 describes the data and the methodology employed. Section 3 deals with the empirical results, while Section 4 contains some concluding comments.
2. Data and methodology The data examined correspond to the log of the CPI, monthly, seasonally unadjusted, for the time period 1970M1–2008M12, obtained from the webpage “Statistics South Africa” (http://www. statssa.gov.za). Though not displayed, the data present an increasing trend over time, suggesting that the log prices are nonstationary. ☆ The author gratefully acknowledges financial support from the Ministerio de Ciencia y Tecnología (ECO2008-03035 ECON Y FINANZAS, Spain) and from a PIUNA Project at the University of Navarra. Comments of an anonymous referee are gratefully acknowledged. ⁎ University of Navarra, Faculty of Economics, Edificio Biblioteca, Entrada Este, E.-31080 Pamplona, Spain. Tel.: +34 948 425 625; fax: +34 948 425 626. E-mail address: [email protected]. 0165-1765/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2011.02.026
In Fig. 1 we display the first 100 sample autocorrelation values on the first differenced data, which corresponds to the inflation process. We observe a slow decrease in the values, which are significant even at lag 50, along with other remarkable values at multiple of 12, implying that the seasonal monthly component may also play a role in describing the behavior of the series. The periodogram of the inflation series is displayed in Fig. 2, and consistently with Fig. 1 we observe large values at zero and the seasonal monthly frequencies. In the empirical results presented in Section 3 we consider the following model, yt = α + βt + xt ;
ð1−LÞd xt = ut ; t = 1; 2; …;
ð1Þ
where α and β are the coefficients corresponding to an intercept and a linear time trend, and xt are the regression errors that are supposed to be I(d) where d can be a real value. The methodology employed to estimate model (1) is based on the Whittle function in the frequency domain (Dahlhaus, 1989) along with a testing procedure developed by Robinson (1994). The latter is a very general method that allows us to consider any real value of d, including thus stationary (d b 0.5) and nonstationary (d ≥ 0.5) hypotheses. Nevertheless, noting that the log prices are clearly nonstationary we also conducted the analysis on the first differenced data, and calling πt = (1 − L)yt we also examined the model, π = β + xt ;
ð1−LÞd xt = ut ;
t = 1; 2; …;
ð2Þ
with I(0) ut, the results being completely consistent with those based on model (1).
208
L.A. Gil-Alana / Economics Letters 111 (2011) 207–209 Table 1 Estimates of d in a fractionally integrated model.
0,4 12
0,3
24
0,2
48
36
60
72
No regressors 84
0,1 0 -0,1
1
An intercept
A linear t ime trend
96
100
Fig. 1. First 100 sample autocorrelation values of the inflation rate. The large sample standard error under the null hypotheses of no autocorrelation is T0.5 or roughly 0.121 for series of length considered here.
0,0003
White noise AR(1) AR(2) Seasonal AR(1) Seasonal AR(2) Bloomfield (1) Bloomfield (2)
0.974 1.355 1.804 0.974 0.989 0.968 1.004
(0.916, (1.256, (1.591, (0.914, (0.906, (0.859, (0.995,
1.043) 1.479) 2.119) 1.044) 1.059) 1.086) 1.103)
1.147 1.261 1.363 1.106 1.103 1.304 1.347
(1.112, (1.213, (1.284, (1.065, (1.060, (1.244, (1.212,
1.189) 1.322) 1.474) 1.155) 1.157) 1.391) 1.499)
1.132 (1.100, 1.171) 1.235 (1.188, 1.297) 1.337 (1.253, 1.465) 1.099 (1.060, 1.146) 1.096 (1.041, 1.153) 1.266 (1.209, 1.355) 1.351 (1.200, 1.677)
In bold is the most appropriate specification based on diagnostic tests on the residuals.
0,1
0,00025
0,05
0,0002 0,00015
0
0,0001
-0,05
0,00005 0
-0,1 1
1
50
T-1
Fig. 2. Periodogram of the inflation rate. The periodogram was calculated based on the discrete Fourier frequencies λj = 2πj/T.
Fig. 3. First 50 sample autocorrelation values of the estimated residuals. The large sample standard error under the null hypotheses of no autocorrelation is T0.5 or roughly 0.121 for series of length considered here.
3. The empirical results
with σε2 = 0.0188, where yt is the log of CPI, and the t-values are displayed in parenthesis. Therefore, a model for inflation is then:
In Table 1 we report the Whittle estimates of d in model (1) along with the 95% confidence bands of the non-rejection values of d using Robinson's (1994) parametric approach, for the three standard cases of no regressors (i.e., α = β = 0 a priori in (1)); an intercept (α unknown and β = 0 a priori); and an intercept with a linear time trend (i.e., α and β unknown). Given the parametric nature of the method employed in this paper we need to specify a functional form for the error term, i.e. ut in Eq. (1). First, we assume that ut is white noise; then, AR(1) and AR(2) processes are imposed; next we consider seasonal monthly AR models, and finally, the exponential spectral model of Bloomfield (1973) is also examined. The latter is a non-parametric approach of modeling the I(0) disturbances that produce autocorrelations decaying exponentially as in the AR(MA) case and that accommodates extremely well in the context of the tests of Robinson (1994). (See, Gil-Alana, 2004). We first observe that if we do not include regressors, the unit root null hypothesis cannot be rejected for the cases of white noise, seasonal AR and Bloomfield disturbances; however if ut is nonseasonal AR, the I(1) model is decisively rejected in favor of higher degrees of integration. On the other hand, including an intercept, or an intercept with a linear trend, the estimated values of d are above 1 in all cases, ranging now from 1.096 in case of seasonal AR(2) ut with a time trend to 1.363 in case of AR(2) ut with an intercept. Performing several diagnostic tests on the estimated residuals the model with seasonal AR(1) disturbances seems to be the most adequate specification for this series.1 That is,
yt = 1:3280 + 0:00792t + xt ; ð1−LÞ ð186:30Þ ð13:579Þ
1:099
xt = ut ;
ð1:060; 1:146Þ
ut = 0:2158ut−12 + εt
1 We performed tests of no serial correlation (Durbin, 1970; Godfrey, 1978a,b); homoscedasticity (Koenker, 1981) and functional form (Ramsey, 1969).
0:099 xt ð0:060; 0:146Þ
πt = ð1−LÞyt = 0:00792 + xt ; ð1−LÞ ð13:579Þ
= ut ; ut = 0:2158ut−12 + εt : 2
The correlogram of the estimated residuals is displayed in Fig. 3 and we observe that practically all values are within the 95% confidence band. Similarly the periodogram (not displayed) presents values relatively flat at the discrete Fourier frequencies, suggesting lack of autocorrelation in this process. This model implies therefore that the inflation in South Africa displays a small, though significant, component of long memory along with another short memory seasonal component. To corroborate the above results we also implemented a semiparametric approach (Robinson, 1995) to estimate the value of d independently of the way of modeling the I (0) error term. We observe, in Fig. 4, that for practically all the bandwidth numbers, displayed in the horizontal axe in the figure, the estimates are above the I(1) interval. This result is consistent with the findings in Rangasamy (2009) who showed that inflation in South Africa was highly persistent by means of using autoregressive models. There are several arguments that can be put forward to explain the long memory feature observed in the South African inflation rate. From a theoretical statistical viewpoint aggregation is the usual argument: Robinson (1978) and Granger (1980) showed that the aggregation of heterogeneous individual AR processes may produce long memory. From an economic viewpoint, imperfect knowledge and information constraints in the economy (Ireland, 2000) and the manner in which nominal contracts are structured (Fuhrer, 2000) are classical arguments used to justify high persistence in inflation rates. Based on the selected model above, Fig. 5 displays the first 60 impulse responses for the inflation series. We observe a fast decrease in the responses with a significant seasonal component that also tends to disappear in the long run.
2 Almost identical results were obtained when directly estimating the parameters with the inflation rate.
L.A. Gil-Alana / Economics Letters 111 (2011) 207–209
convergence process to the mean is hyperbolic though disappearing relatively fast.
2 1,5 1
References
0,5 0
1
T/2
Fig. 4. Estimates of d based on the semiparametric Whittle method of Robinson (1995). The horizontal axe refers to the bandwidth parameter number, while the vertical one displays the estimates of d. The thin lines refer to the 95% confidence band for the I(1) case.
0,4 0,3 0,2 0,1 0
209
1
12
24
36
48
60
Fig. 5. First 60 impulse response values.
4. Concluding comments In this paper we have examined the structure of the log-CPI in South Africa by using fractional integration. In doing so we permit a greater degree of flexibility in the dynamic specification of the series than the standard methods based on I(0) stationarity and I(1) nonstationarity. Our results indicate that inflation in South Africa can be well described in terms of an I(d) process with seasonal AR disturbances. The estimated values of d appear to be positive and small though significantly different from zero, thus implying that the
Backus, D., Zin, S., 1993. Long memory inflation uncertainty. Evidence from the term structure of interest rates. Journal of Money, Credit and Banking 25, 681–700. Bloomfield, P., 1973. An exponential model in the spectrum of a scalar time series. Biometrika 60, 217–226. Dahlhaus, R., 1989. Efficient parameter estimation for self-similar process. Annals of Statistics 17, 1749–1766. Delgado, M., Robinson, P.M., 1994. New methods for the analysis of long memory time series. Application to Spanish inflation. Journal of Forecasting 13, 97–107. Durbin, J., 1970. Testing for serial correlation in least squares regression when some of the regressors are lagged dependent variables. Econometrica 38, 410–421. Fuhrer, J., 2000. Habit formation in consumption and its implications for monetary policy models. The American Economic Review 90, 367–390. Gil-Alana, L.A., 2004. The use of the Bloomfield (1973) model as an approximation to ARMA processes in the context of fractional integration. Mathematical and Computer Modelling 39, 429–436. Godfrey, L.G., 1978a. Testing against general autoregressive and moving average models when the regressors include lagged dependent variables. Econometrica 43, 1293–1301. Godfrey, L.G., 1978b. Testing for higher order serial correlation in regression equations when the regressors include lagged dependent variables. Econometrica 43, 1303–1310. Granger, C.W.J., 1980. Long memory relationships and the aggregation of dynamic models. Journal of Econometrics 14, 227–238. Hassler, U., 1993. Regression of spectral estimators with fractionally integrated time series. Journal of Time Series Analysis 14, 369–380. Ireland, P., 2000. Expectations, credibility and time consistent monetary policy. Macroeconomic Dynamics 4, 448–466. Koenker, R., 1981. A note of studentizing a test for heteroscedasticity. Journal of Econometrics 17, 107–112. Ramsey, J.B., 1969. Tests for specification errors in classical linear least squares regression analysis. Journal of the Royal Statistical Society, Series B 31, 350–371. Rangasamy, L., 2009. Inflation persistence and core inflation. The case of South Africa. South African Journal of Economics 77, 430–444. Robinson, P.M., 1978. Statistical inference for a random coefficient autoregressive model. Scandinavian Journal of Statistics 5, 163–168. Robinson, P.M., 1994. Efficient tests of nonstationary hypotheses. Journal of the American Statistical Association 89, 1420–1437. Robinson, P.M., 1995. Gaussian semiparametric estimation of long range dependence. Annals of Statistics 23, 1630–1661.
Contents lists available at ScienceDirect
Economics Letters j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e c o l e t
Inflation in South Africa. A long memory approach☆ Luis A. Gil-Alana ⁎ University of Navarra, Faculty of Economics, Pamplona, Spain
a r t i c l e
i n f o
Article history: Received 25 March 2010 Received in revised form 20 January 2011 Accepted 15 February 2011 Available online 21 February 2011
a b s t r a c t This paper deals with the analysis of the inflation rate in South Africa for the time period 1970M1–2008M12. We use long range dependence techniques and the results show that inflation in this country is a covariance stationary process with long range dependence, with an order of integration ranging in the interval (0, 0.5). Policy implications are derived. © 2011 Elsevier B.V. All rights reserved.
JEL classification: C22 Keywords: Inflation South Africa Long range dependence
1. Introduction Long range dependence or long memory is a characteristic of many economic time series including inflation rates. (See, for example, the paper of Backus and Zin, 1993, for the US case; and Hassler, 1993, and Delgado and Robinson, 1994, for the Swiss and Spanish inflation rates respectively). In this paper we use fractional integration to show that the inflation rate in South Africa may also be characterized as an I(d) process with the value of d ranging in the interval (0, 0.5). Thus, inflation in this country may be well described as a covariance stationary process with long range dependence. The structure of the article is as follows: Section 2 describes the data and the methodology employed. Section 3 deals with the empirical results, while Section 4 contains some concluding comments.
2. Data and methodology The data examined correspond to the log of the CPI, monthly, seasonally unadjusted, for the time period 1970M1–2008M12, obtained from the webpage “Statistics South Africa” (http://www. statssa.gov.za). Though not displayed, the data present an increasing trend over time, suggesting that the log prices are nonstationary. ☆ The author gratefully acknowledges financial support from the Ministerio de Ciencia y Tecnología (ECO2008-03035 ECON Y FINANZAS, Spain) and from a PIUNA Project at the University of Navarra. Comments of an anonymous referee are gratefully acknowledged. ⁎ University of Navarra, Faculty of Economics, Edificio Biblioteca, Entrada Este, E.-31080 Pamplona, Spain. Tel.: +34 948 425 625; fax: +34 948 425 626. E-mail address: [email protected]. 0165-1765/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2011.02.026
In Fig. 1 we display the first 100 sample autocorrelation values on the first differenced data, which corresponds to the inflation process. We observe a slow decrease in the values, which are significant even at lag 50, along with other remarkable values at multiple of 12, implying that the seasonal monthly component may also play a role in describing the behavior of the series. The periodogram of the inflation series is displayed in Fig. 2, and consistently with Fig. 1 we observe large values at zero and the seasonal monthly frequencies. In the empirical results presented in Section 3 we consider the following model, yt = α + βt + xt ;
ð1−LÞd xt = ut ; t = 1; 2; …;
ð1Þ
where α and β are the coefficients corresponding to an intercept and a linear time trend, and xt are the regression errors that are supposed to be I(d) where d can be a real value. The methodology employed to estimate model (1) is based on the Whittle function in the frequency domain (Dahlhaus, 1989) along with a testing procedure developed by Robinson (1994). The latter is a very general method that allows us to consider any real value of d, including thus stationary (d b 0.5) and nonstationary (d ≥ 0.5) hypotheses. Nevertheless, noting that the log prices are clearly nonstationary we also conducted the analysis on the first differenced data, and calling πt = (1 − L)yt we also examined the model, π = β + xt ;
ð1−LÞd xt = ut ;
t = 1; 2; …;
ð2Þ
with I(0) ut, the results being completely consistent with those based on model (1).
208
L.A. Gil-Alana / Economics Letters 111 (2011) 207–209 Table 1 Estimates of d in a fractionally integrated model.
0,4 12
0,3
24
0,2
48
36
60
72
No regressors 84
0,1 0 -0,1
1
An intercept
A linear t ime trend
96
100
Fig. 1. First 100 sample autocorrelation values of the inflation rate. The large sample standard error under the null hypotheses of no autocorrelation is T0.5 or roughly 0.121 for series of length considered here.
0,0003
White noise AR(1) AR(2) Seasonal AR(1) Seasonal AR(2) Bloomfield (1) Bloomfield (2)
0.974 1.355 1.804 0.974 0.989 0.968 1.004
(0.916, (1.256, (1.591, (0.914, (0.906, (0.859, (0.995,
1.043) 1.479) 2.119) 1.044) 1.059) 1.086) 1.103)
1.147 1.261 1.363 1.106 1.103 1.304 1.347
(1.112, (1.213, (1.284, (1.065, (1.060, (1.244, (1.212,
1.189) 1.322) 1.474) 1.155) 1.157) 1.391) 1.499)
1.132 (1.100, 1.171) 1.235 (1.188, 1.297) 1.337 (1.253, 1.465) 1.099 (1.060, 1.146) 1.096 (1.041, 1.153) 1.266 (1.209, 1.355) 1.351 (1.200, 1.677)
In bold is the most appropriate specification based on diagnostic tests on the residuals.
0,1
0,00025
0,05
0,0002 0,00015
0
0,0001
-0,05
0,00005 0
-0,1 1
1
50
T-1
Fig. 2. Periodogram of the inflation rate. The periodogram was calculated based on the discrete Fourier frequencies λj = 2πj/T.
Fig. 3. First 50 sample autocorrelation values of the estimated residuals. The large sample standard error under the null hypotheses of no autocorrelation is T0.5 or roughly 0.121 for series of length considered here.
3. The empirical results
with σε2 = 0.0188, where yt is the log of CPI, and the t-values are displayed in parenthesis. Therefore, a model for inflation is then:
In Table 1 we report the Whittle estimates of d in model (1) along with the 95% confidence bands of the non-rejection values of d using Robinson's (1994) parametric approach, for the three standard cases of no regressors (i.e., α = β = 0 a priori in (1)); an intercept (α unknown and β = 0 a priori); and an intercept with a linear time trend (i.e., α and β unknown). Given the parametric nature of the method employed in this paper we need to specify a functional form for the error term, i.e. ut in Eq. (1). First, we assume that ut is white noise; then, AR(1) and AR(2) processes are imposed; next we consider seasonal monthly AR models, and finally, the exponential spectral model of Bloomfield (1973) is also examined. The latter is a non-parametric approach of modeling the I(0) disturbances that produce autocorrelations decaying exponentially as in the AR(MA) case and that accommodates extremely well in the context of the tests of Robinson (1994). (See, Gil-Alana, 2004). We first observe that if we do not include regressors, the unit root null hypothesis cannot be rejected for the cases of white noise, seasonal AR and Bloomfield disturbances; however if ut is nonseasonal AR, the I(1) model is decisively rejected in favor of higher degrees of integration. On the other hand, including an intercept, or an intercept with a linear trend, the estimated values of d are above 1 in all cases, ranging now from 1.096 in case of seasonal AR(2) ut with a time trend to 1.363 in case of AR(2) ut with an intercept. Performing several diagnostic tests on the estimated residuals the model with seasonal AR(1) disturbances seems to be the most adequate specification for this series.1 That is,
yt = 1:3280 + 0:00792t + xt ; ð1−LÞ ð186:30Þ ð13:579Þ
1:099
xt = ut ;
ð1:060; 1:146Þ
ut = 0:2158ut−12 + εt
1 We performed tests of no serial correlation (Durbin, 1970; Godfrey, 1978a,b); homoscedasticity (Koenker, 1981) and functional form (Ramsey, 1969).
0:099 xt ð0:060; 0:146Þ
πt = ð1−LÞyt = 0:00792 + xt ; ð1−LÞ ð13:579Þ
= ut ; ut = 0:2158ut−12 + εt : 2
The correlogram of the estimated residuals is displayed in Fig. 3 and we observe that practically all values are within the 95% confidence band. Similarly the periodogram (not displayed) presents values relatively flat at the discrete Fourier frequencies, suggesting lack of autocorrelation in this process. This model implies therefore that the inflation in South Africa displays a small, though significant, component of long memory along with another short memory seasonal component. To corroborate the above results we also implemented a semiparametric approach (Robinson, 1995) to estimate the value of d independently of the way of modeling the I (0) error term. We observe, in Fig. 4, that for practically all the bandwidth numbers, displayed in the horizontal axe in the figure, the estimates are above the I(1) interval. This result is consistent with the findings in Rangasamy (2009) who showed that inflation in South Africa was highly persistent by means of using autoregressive models. There are several arguments that can be put forward to explain the long memory feature observed in the South African inflation rate. From a theoretical statistical viewpoint aggregation is the usual argument: Robinson (1978) and Granger (1980) showed that the aggregation of heterogeneous individual AR processes may produce long memory. From an economic viewpoint, imperfect knowledge and information constraints in the economy (Ireland, 2000) and the manner in which nominal contracts are structured (Fuhrer, 2000) are classical arguments used to justify high persistence in inflation rates. Based on the selected model above, Fig. 5 displays the first 60 impulse responses for the inflation series. We observe a fast decrease in the responses with a significant seasonal component that also tends to disappear in the long run.
2 Almost identical results were obtained when directly estimating the parameters with the inflation rate.
L.A. Gil-Alana / Economics Letters 111 (2011) 207–209
convergence process to the mean is hyperbolic though disappearing relatively fast.
2 1,5 1
References
0,5 0
1
T/2
Fig. 4. Estimates of d based on the semiparametric Whittle method of Robinson (1995). The horizontal axe refers to the bandwidth parameter number, while the vertical one displays the estimates of d. The thin lines refer to the 95% confidence band for the I(1) case.
0,4 0,3 0,2 0,1 0
209
1
12
24
36
48
60
Fig. 5. First 60 impulse response values.
4. Concluding comments In this paper we have examined the structure of the log-CPI in South Africa by using fractional integration. In doing so we permit a greater degree of flexibility in the dynamic specification of the series than the standard methods based on I(0) stationarity and I(1) nonstationarity. Our results indicate that inflation in South Africa can be well described in terms of an I(d) process with seasonal AR disturbances. The estimated values of d appear to be positive and small though significantly different from zero, thus implying that the
Backus, D., Zin, S., 1993. Long memory inflation uncertainty. Evidence from the term structure of interest rates. Journal of Money, Credit and Banking 25, 681–700. Bloomfield, P., 1973. An exponential model in the spectrum of a scalar time series. Biometrika 60, 217–226. Dahlhaus, R., 1989. Efficient parameter estimation for self-similar process. Annals of Statistics 17, 1749–1766. Delgado, M., Robinson, P.M., 1994. New methods for the analysis of long memory time series. Application to Spanish inflation. Journal of Forecasting 13, 97–107. Durbin, J., 1970. Testing for serial correlation in least squares regression when some of the regressors are lagged dependent variables. Econometrica 38, 410–421. Fuhrer, J., 2000. Habit formation in consumption and its implications for monetary policy models. The American Economic Review 90, 367–390. Gil-Alana, L.A., 2004. The use of the Bloomfield (1973) model as an approximation to ARMA processes in the context of fractional integration. Mathematical and Computer Modelling 39, 429–436. Godfrey, L.G., 1978a. Testing against general autoregressive and moving average models when the regressors include lagged dependent variables. Econometrica 43, 1293–1301. Godfrey, L.G., 1978b. Testing for higher order serial correlation in regression equations when the regressors include lagged dependent variables. Econometrica 43, 1303–1310. Granger, C.W.J., 1980. Long memory relationships and the aggregation of dynamic models. Journal of Econometrics 14, 227–238. Hassler, U., 1993. Regression of spectral estimators with fractionally integrated time series. Journal of Time Series Analysis 14, 369–380. Ireland, P., 2000. Expectations, credibility and time consistent monetary policy. Macroeconomic Dynamics 4, 448–466. Koenker, R., 1981. A note of studentizing a test for heteroscedasticity. Journal of Econometrics 17, 107–112. Ramsey, J.B., 1969. Tests for specification errors in classical linear least squares regression analysis. Journal of the Royal Statistical Society, Series B 31, 350–371. Rangasamy, L., 2009. Inflation persistence and core inflation. The case of South Africa. South African Journal of Economics 77, 430–444. Robinson, P.M., 1978. Statistical inference for a random coefficient autoregressive model. Scandinavian Journal of Statistics 5, 163–168. Robinson, P.M., 1994. Efficient tests of nonstationary hypotheses. Journal of the American Statistical Association 89, 1420–1437. Robinson, P.M., 1995. Gaussian semiparametric estimation of long range dependence. Annals of Statistics 23, 1630–1661.