Hamilton J. D. (1994). Time Series Analysis. Princeton University Press

[Saundra Balock]

Animportant limitation in order to specify and estimate a macroeconomic model that describes the Chilean economy resides in using variables with sufficient number of observations that allow for a reliable econometric estimation. Among these variables, the GDP constitutes a fundamental magnitude. Nevertheless, for this variable there is not quarterly information before 1980. This paper computes quarterly GDP series for the period 1965-1980 using the approach by Casals et al. (2000). As result, the new series incorporates the cyclical dynamic in the quarterly series later to 1979 respecting, in addition, all the annual existing information before the above mentioned period.

Una importante limitacin a la hora de especificar y estimar un modelo macroeconmico que describa la economa chilena reside en utilizar variables con suficientes observaciones que permitan una estimacin economtrica fiable. Entre ellas, el PIB constituye una magnitud fundamental. Sin embargo, no existe informacin trimestral previa a 1980. En este trabajo se construye la serie trimestral del PIB chileno para el perodo 1966-1979 utilizando la metodologa de interpolacin propuesta por Casals et al. (2000). Como resultado, la nueva serie incorpora la dinmica cclica en las series trimestrales posteriores a 1979 respetando, adems, toda la informacin anual existente antes de dicho perodo.

An important limitation in order to specify and estimate a macroeconomic model that describes the Chilean economy resides in using variables with sufficient number of observations that allow for reliable econometric information. Among these variables, the GDP constitutes a fundamental magnitude. However, although there is annual information of this series from the XIX century, quarterly information is only available from 1980.

In this paper we reconstruct quarterly series of the Chilean GDP for the period 1966-1979 using the interpolation methodology advocated by De Jong (1989) and Casals et al. (2000) for the nonstationary case. This procedure, explained in a simple way, consists firstly of: 1) specifying and estimating a statistical model of the series of interest; 2) expressing the resulting model in state space form; and then 3) reconstructing this series backward using a fixed-interval smoothing algorithm.

As far as we are aware, in spite of its empirical relevance, there is only a previous attempt of reconstructing the Chilean GDP by Haindl (1986). An important difference between his paper and ours is that he uses the methodology proposed by Chow and Lin (1971). This methodology imposes a very restrictive functional form that cannot represent accurately the features of our data. More specifically, this procedure assumes that: 1) the dependent variable and the set of indicators are fully cointegrated; 2) there is not feedback between the variables in the model; and 3) the residual of the regression between the dependent variable and the set of indicators follows an ad-hoc functional form.

Here, instead of imposing any particular functional form on the variables, we specify models that are plausible given the statistical properties of the time series in our analysis. Two alternative statistical specifications are considered in our analysis: a univariate ARIMA model and a transfer function model. These two specifications are robust and filter the data to uncorrelated and normal residuals. The quarterly series obtained under both procedures incorporate the cyclical dynamic in the quarterly series later to 1979 respecting, additionally, all the previous annual information.

The remainder of the paper is organized as follows. In the next section, we specify and estimate an ARIMA model for the Chilean GDP. Section 3 discusses how to write the univariate model in the state space form and how to use this formulation to reconstruct the quarterly series backward using a fixed interval smoothing algorithm. In Section 4, we extend the analysis to consider the use of transfer function models. Quarterly GDP series obtained from the univariate and the transfer function model are shown and compared in Section 5. Some concluding remarks follow in Section 6.

In this section we describe some of the important features of the individual time series used in the analysis and then we specify and estimate an ARIMA model for the Chilean GDP that will be used to interpolate quarterly values of this series before 1980. Our main interest is in Chilean GDP in real terms. This series is freely available from the Central Bank of Chile at the following URL: The Central Bank of Chile publishes GDP series on a quarterly basis since 1980. However, it is possible to find GDP information on annual basis before this period.

Additionally, we also consider other quarterly series that will be used in the subsequent analysis to specify an econometric model. More specifically, our intention is to study the properties of these series to use them as indicators to interpolate quarterly information of the Chilean GDP before 1980. Three series have received our attention in this respect: 1) the monetary aggregate M1; 2) the price of copper; and 3) the terms of trade.

Series of the monetary aggregate M1 were collected from the Monthly Bulletin of the Central Bank of Chile. Series of the price of copper was obtained from "Informe Econmico y Financiero'', also published by the Central Bank of Chile. Series of the terms of trade can be found in the paper by Bennett and Valds (2001). These three series are available from 1966.

In order to specify an univariate ARIMA model for the GDP series we choose not to consider the first 10 quarterly observations as they are very erratic and generate an spurious AR(1) or AR(2) structure. That is, we only consider the period 1982:3-2004:2. The evolution of this series is shown in Figure 1. Inspection of the figure reveals that the GDP grows during the period under consideration and this growth is affected by the seasonal cycle.

Figures of the monetary aggregate M1, the price of copper and the terms of trade are not shown for the sake of brevity. However, M1 grows during the period and its growth is also affected by the seasonal cycle. Series of the terms of trade and the price of copper, on the other hand, do not grow but show little tendency to return to mean.

This and the additional information provided by correlograms suggests that M1 and GDP series require two differences to become stationary and one of them could be a seasonal difference. Also, terms of trade and price of copper should be stationary after one regular difference.

More formally, we employ the Augmented Dickey-Fuller (ADF) test for unit roots on the series in levels, first and second differences. It is, of course, necessary to choose the number of augmentation lags to account for serial correlation and this is done using the sequential approach in Ng and Perron (1995). The results are shown in Table 1.

For series in levels, the unit root null hypothesis cannot be rejected at conventional significance levels in any case. However, the unit root null can be rejected at the 0.01 level for first differences of the price of copper and terms of trade. For M1 and GDP it is necessary to take second differences in order to reject the null at the 0.01 level.

Once the number of unit roots has been determined for each of the series, the next step is to specify and estimate an ARIMA model for the Chilean GDP. In order to do this, we use the Box-Jenkins methodology based on the observation and interpretation of correlograms; see Box et al. (1994). The simple correlogram for the GDP series with one regular and one seasonal difference show that the series is stationary and it does not have any known structure in the regular part. However, the seasonal part suggests either a MA(1) or AR(2) specification. When the two models are estimated, the standard deviation of the estimated residuals is 0.12 in both cases. However, a MA(1) is more parsimonious than an AR(2) model and we choose the first option. This amounts to specifying the following model for the Chilean GDP (denoted by yt):

The correlation between the two estimated parameters (σ and ς) is always below 0.01. Besides, the Jarque-Bera statistic on the estimated residuals is 1.80 indicates that they can be considered as normal. Also, the correlogram of the residuals do not show any significant correlation at any peak and its structure could be regarded as a white noise process.

Once we have defined a model for Yt, the quarterly series is constructed using a fixed interval smoothing algorithm where for each four values one is fixed. For a brief description of this process, notice that every linear econometric model with fixed parameters can be represented in the state space form as:2

A fixed interval smoothing algorithm consists on obtaining an estimation of the state variable and its variance from the available sample information. We denote these estimations as . 3 A detailed description of the computational procedure of interpolation proposed by De Jong (1989) and extended by Casals et al. (2000) is confined to the appendix. Intuitively, the algorithm departs from some initial conditions α1 and P1 and then estimates the estate variable using a Kalman filter and reconstructs the series backward using a smoothing algorithm.

An important drawback in the algorithm advocated by De Jong (1989) lies in the fact that the initial value of the covariance of the state value, P1, is arbitrarily close to infinite for all the states. Casals et al. (2000) extend this previous methodology by proposing a fixed interval algorithm that allows for both, stationary and unit roots, and also treats the case of exogenous inputs. This last procedure is used here to interpolate nonexistent values of yt. Thus, from 1980 backward, for each 4 values, 3 nonexistent values are interpolated using E4.4 The algorithm gives the interpolated values of yt and the estimated standard deviations of the interpolations.

3a8082e126