Describing the Phillips-curve with a regression model faces two major problems. First, as is common for time series data, there exists a high positive serial correlation in the residuals of the estimated regression model. Second, time series of income changes and unemployment rates may be non-stationary, that is, both series may have been generated by a stochastic (or random) process. If this is the case, then ordinary least squares (OLS) would not yield a consistent parameter estimator.
The results in this paper show that income changes and unemployment rates are indeed non-stationary time series both at the national (Austria) and at the regional level (Austria's nine states) for the period between 1967-1997. This paper further proceeds with co-integration tests, using the methodology developed by Johansen (1995), to find out whether or not a linear combination of the two non-stationary time series is stationary. The results of the co-integration analysis show that stationary linear combinations between both time series exist for Austria and its nine states. These linear combinations are expressed in the form of co-integration equations, which may be interpreted as the long-run equilibrium relationship between the rate of income change and the unemployment rate. The main conclusion of this paper is that co-integration rather than regression analysis should be used to analyze and interpret the relationship expressed in the Phillips-curve.
In Section 2 of this article the income and unemployment data used in the analysis are introduced. Regression models in the form of the classical Phillips-curve are estimated for Austria and its nine states in Section 3. It turns out that expressing the relationship between the rate of income change and the unemployment rate in the form of the Phillips-curve has two major problems. First, there exists high positive serial correlation in the residuals of the estimated equations and second, the rate of income change and the (inverse) unemployment rate time series are non-stationary. Regressing one non-stationary time series against another can lead to spurious regression results. To overcome both problems, a co-integration analysis is performed on the same data. In Section 4, the augmented Dickey-Fuller (ADF) and the Phillips-Perron (PP) unit root tests are applied to find out whether the rate of income change and the inverse unemployment rate time series follow a non-stationary process. The results of the unit root tests show that the time series are indeed non-stationary and integrated at order one. Co-integration equations, relating the rate of income change with the inverse unemployment rate, are estimated in Section 5. The co-integration analysis is performed with a vector autoregression (VAR)-based co-integration test, using the methodology developed by Johansen (1995). The final section summarizes the results of this research. The entire data analysis was carried out with EViews, vers. 3.1 (Quantitative Micro Software, 1998a), an econometric software package, primarily developed to investigate time series data.
'When the demand for labor is high and there are very few unemployed we should expect employers to bid wage rates up quite rapidly, each firm and each industry being continually tempted to offer a little above the prevailing rates to attract the most suitable labor from other firms and industries. On the other hand it appears that workers are reluctant to offer their services at less than the prevailing rates when the demand for labor is low and unemployment is high so that wage rates fall only very slowly. The relation between unemployment and the rate of change of wage rates is therefore likely to be highly non-linear.'
Phillips (1958) used the index of hourly wage rates to calculate the rate of change of money wage rates by expressing half the first central difference of the index for each year as a percentage of the index for the same year. Half the first central difference is thus intended to measure the average of wage rates during each year. In contrast to Phillips (1958), the median monthly gross salary (per year) is used in this article to calculate the rate of change of money wage rates with half the first central difference as follows:
Thus the rate of income change for 1967 (RINC1967) is taken to be half the difference between the income for 1968 (INC1968) and the income for 1966 (INC1966) expressed as a percentage of the income for 1967 (INC1967). All subsequent analyses are based on 31 yearly rates of income change and yearly unemployment data from 1967 through 1997 for Austria and its nine states.
Phillips-curves for Austria and its nine states are shown in Figure 1. The unemployment rate is plotted on the x-axis, while the rate of income change is plotted on the y-axis. 1967 denotes the starting year (i.e., the beginning of the Phillips-curve), 1997 marks the final year of the observation period (i.e., the ending of the Phillips-curve). In between, years (small black points) are connected consecutively with straight lines. With maybe the exception of the Burgenland, there is a clear tendency for the rate of income change to be high when unemployment rate is low and for the rate of income change to be low when the unemployment rate is high. In general, the period between the end of the 60's and the mid-70's is marked by increasing rates of income changes and decreasing unemployment rates. Since the mid-70's, the Phillips-curves show decreasing rates of income changes and increasing unemployment rates. With the exceptions of Carinthia, Styria, and Upper Austria 1997 exhibits the lowest rate of income change paired with the highest unemployment rate throughout the entire observation period.
Table 1 includes the results of the non-linear regression model documenting the relationship between the rate of income change (RINC) and the unemployment rate (UNEMPR). The model being tested has the form:
This regression model is the same model, originally tested by Phillips (1958) for the U. K. This followed an enormous flood of research articles, in which the relationship between the income change and the unemployment rate was repeatedly tested in different spatial and/or temporal contexts and/or with the inclusion of further explanatory variables (i.e., consumer price index, number of vacancies, %-unionization, etc.) into the original regression model (Lipsey, 1960; Thirlwall, 1970). In Austria, Breuss (1980) and Wörgötter (1975, 1978) tested this model with different income data and observation periods at the national level. Leitner (1990) estimated the same model not only for Austria, but also for its nine states.
All ten regression models in Table 1 are significant at the 1%-level. Austria possesses the best model fit (86% explained), Burgenland the lowest (40% explained). The regression coefficient 'b' is significant at 1% for all regions, 'a' is significant at least at the 5%-level in 6 out of the 10 regions. The low positive Durbin-Watson (DW) statistic indicates positive serial correlation in the residuals of the estimated equation that is, residuals correlate positively with their own lagged values. Serial correlation is a common finding in time series regression. It violates the standard assumption of regression theory that disturbances are not correlated with other disturbances. The existence of serial correlation in the residuals will lead to incorrect estimates of the standard errors, and invalid statistical inference (e.g., hypothesis tests and forecasting) for the coefficients of the equation.
The DW statistic is a test for first-order serial correlation that is, it measures the linear association between adjacent residuals from a regression model. A DW statistic around 2 would indicate no serial correlation. In Table 1 the DW statistic ranges from a low value of 0.458 (Tyrol) to a high value of 1.383 (Carinthia). In order to verify the results of the DW statistic, the Breusch-Godfrey test for first-order serial correlation in the residuals was also performed for all ten regions. The individual results of the Breusch-Godfrey test are not shown here, but, in general, they confirm the results of the DW statistic. The only exception is Carinthia, for which the Breusch-Godfrey test fails to reject the null hypothesis of no first-order serial correlation in the residuals. Carinthia also showed the highest value for the DW statistic in Table 1.
A simple example of a stochastic time series is the random walk process. Each successive change in yT is drawn independently from a probability distribution with 0 means. Thus yT is determined by
A simple extension of the random walk process is the random walk with drift. This process accounts for a trend (upward or downward) in the series yT and thereby allows to embody that trend in the forecast. In this series, yT is determined by
so that on the average the process will tend to move upward (for d > 0) (Pindyck and Rubinfeld, 1991).
If the underlying stochastic process that generated the series can be assumed to be invariant with respect to time, the process or series is non-stationary. If the stochastic process is fixed in time, the process or series is stationary. Stationary series can be modeled via an equation with fixed coefficients that can be estimated from past data. The random walk with drift is one example of a non-stationary process. Many of the time series encountered in business and economics are not generated by stationary processes. The GNP, for example, has for the most part been growing steadily, and for this reason alone its stochastic properties (e.g., mean) in 1980 are different from those in 1933 (Nelson and Plosser, 1982; Pindyck and Rubinfeld, 1991).
Many of the non-stationary time series encountered have the desirable property that if they are differenced one or more times, the resulting series will be stationary. Such a non-stationary series is termed homogenous. The number of times that the original series must be differenced before a stationary series results is called the order of homogeneity. Thus, if yt is first-order homogeneous non-stationary, the series
is stationary. If yt happened to be second-order homogenous, the series
would be stationary (Pindyck and Rubinfeld, 1991).
A difference stationary series is said to be integrated and is denoted as I(d) where d is the order of integration. The order of integration is the number of unit roots contained in the series, or the number of differencing operations it takes to make the series stationary. The random walk process, discussed above, contains one unit root, so it is an I(1) series. Similarly, a stationary series is I(0).
The above discussion is important insofar as a regression of one non-stationary time series against another can lead to spurious results, because ordinary least squares (OLS) would not yield a consistent parameter estimator. Therefore, it is important to check whether a series is stationary or not before using it in a regression. The formal method to test the stationarity of a series is the unit root test.
During the remainder of this section, the rate of income change (RINC) and the inverse unemployment rate (IUNEMPR) time series for Austria and its nine states will be tested whether they are stationary or not. The inverse unemployment rate is used instead of the unemployment rate, because the IUNEMPR time series for most regions show a remarkable similarity with the time series for the rate of income change (compare Figure 2). Also, the regression model estimated in Section 3 basically relates the rate of income change with the IUNEMPR (because the parameter c is set to -1).
Two widely used unit root tests will be applied to the two time series data for Austria and its nine states: the augmented Dickey-Fuller (ADF) test (Dickey and Fuller, 1979) and the Phillips-Perron (PP) test (Phillips and Perron, 1988). Both tests control for higher-order serial correlation in the series. The ADF approach controls for higher-order correlation by adding lagged difference terms of the dependent variable y to the right-hand side of the regression (Quantitative Micro Software, 1998b):
The PP test makes a correction to the t-statistic of the ( coefficient from the first-order autoregressive model to account for the serial correlation in ( (Quantitative Micro Software, 1998b).
Figure 2 shows the rate of income change (RINC) and the inverse unemployment rate (IUNEMPR) time series for Austria and its nine states. In general, the time series follow a downward trend from the beginning of the 1970's to the end of the observation period. This means that the stochastic processes that generated the series can be assumed to be invariant with respect to time. When comparing the two individual time series (RINC and IUNEMPR) for one region, one can observe a remarkable similarity. This very instance is important for the co-integration analysis, discussed in the following section.
Table 2 shows the results of the ADF and PP unit root tests for the rate of income change and the inverse unemployment rate for Austria and its nine states. For each time series, both the ADF and the PP tests were run three different times: first, neither a constant nor a trend was included in the test regression (this assumes that the series fluctuates around a zero mean); second, a constant was included (this assumes that the series does not exhibit any trend and has a nonzero mean); third, a constant and a trend was included (this assumes that the series contains a trend). Also, the number of lagged first difference terms (in case of the ADF test) and the number of periods of serial correlation to include in the test regression (in case of the PP test) was determined for each time series. These values were taken from correlograms that were derived for each series and included in Table 2 (values in parenthesis). A '1' in Table 2 indicates that the series is integrated at order one (i.e., has one unit root), a '2' means that the series is integrated at order two (i.e., has two unit roots), and a '0' denotes that the series is stationary. The majority of the time series tested are statistically significant non-stationary with one unit root. The results are very consistent for both the ADF- and the PP-test for test regressions including neither a constant nor a trend or including only a constant, but are inconclusive for the third test category. It follows from the results that the unit root test regression could include a constant, but need not to include one. In either cases, the time series are non-stationary and integrated at order one.
As mentioned above, regressing one non-stationary time series against another can lead to spurious results, in that significant tests will tend to indicate a relationship between the variables when in fact non exists. If a test fails to reject the hypothesis of non-stationarity, one can difference the series in question before using it in a regression. While this is acceptable, differencing may result in a loss of information about the long-run relationship between two variables (Pindyck and Rubinfeld, 1991). A better solution would be to run a co-integration test. This will be discussed in the following section.
Co-integration literature goes back to Yule (1926), who suggested that regressions based on trending time series data can be spurious. This problem of spurious regression was further pursued by Granger and Newbold (1974) and this also led to the concept of co-integration. The literature on co-integration has finally exploded after the pathbreaking paper by Granger (1981). The earliest co-integration test was the one suggested in Engle and Granger (1987) which consists of estimating the co-integrating regression by OLS, obtaining the (estimated) residuals ut and applying unit root tests for ut. Several extensions of this test have been proposed and since they are all based on ut, they are called residual-based tests.
Following, the co-integration analysis is performed with a vector autoregression (VAR)-based co-integration test, using the methodology developed by Johansen (1995). Johansen's method is to test the restrictions imposed by co-integration on the unrestricted VAR involving the series (Quantitative Micro Software, 1998b).
The following co-integration equations express the estimated co-integrating relationship between the rate of income change (RINC) and the inverse unemployment rate (IUNEMPR) for Austria and its nine states. These relationships provide an estimate of the long run propensity to the rate of income change. All ten equations were estimated under the assumption that the series have linear trends but the co-integrating equations have only intercepts.
Co-integration equation for Austria: RINC - 12.62405*IUNEMPR - 1.771069
Co-integration equation for Burgenland: RINC - 52.29852* IUNEMPR + 3.082215
Co-integration equation for Carinthia: RINC - 43.93123* IUNEMPR + 1.610354
Co-integration equation for Lower Austria: RINC - 14.38005* IUNEMPR - 1.353146
Co-integration equation for Salzburg: RINC - 14.05800* IUNEMPR + 0.334111
Co-integration equation for Styria: RINC - 15.74614* IUNEMPR - 1.838410
Co-integration equation for Tyrol: RINC - 13.05425* IUNEMPR - 1.635564
Co-integration equation for Upper Austria: RINC - 11.23605* IUNEMPR - 1.651676
Co-integration equation for Vienna: RINC - 6.656216* IUNEMPR - 2.826227
Co-integration equation for Vorarlberg: RINC - 2.794127* IUNEMPR - 3.635067
Figure 3 shows the co-integrating relations for Austria and its nine states. The zero line in each graph denotes the long run equilibrium of the rate of income change based on the unemployment rate. From the beginning of the observation period until the early 70's, the rate of income change lies above the long run equilibrium for most regions, with the exceptions of Salzburg, Vienna, and Vorarlberg, where the rate is below. From the mid-70's until the early 80's, increases in income rates were less than expected in all regions. The general trend for the next decade (beginning 80's until the beginning of the 90's) shows above average income increases. Since then, the rate of income change has decreased steadily and has again fallen below the long run level around the mid-90's. In 50% of all regions (i.e., Austria, Lower Austria, Tyrol, Vienna, and Vorarlberg), the final year of the observation period (1997) is marked by the smallest income increase with respect to the long run equilibrium.
Co-integration analysis requires the time series under investigation to be non-stationary. Two unit root tests, the augmented Dickey-Fuller (ADF) and the Phillips-Perron (PP) tests were applied to find out whether the rate of income change and the inverse unemployment rate time series are non-stationary. The results show that the time-series for all ten regions are non-stationary and integrated at order one for test regressions that do not include a trend but may include a constant.
This paper reports on work in progress. Future research will concentrate on the inclusion of additional explanatory variables (e.g., consumer price index, number of vacancies, %-unionization, human capital, etc.) into the unit root and co-integration analysis. It will be hypothesized that such variables relate positively to the rate of income change. In addition, the number of observations for the existing data set (and for the additional variables) will be increased by including observations before 1966 and/or by decreasing the interval between observations from one year to, for example, half a year. If such data are available at the regional (state) level is, however, questionable. Additional research should further explore the co-integration relationship between the unemployment and the explanatory variables discussed in this paper. There is much need to develop alternative co-integration models in addition to the prototype suggested in this paper. This may be achieved by changing the assumptions (i.e., inclusion of intercepts and/or trends) made and the lag intervals of the series and/or the co-integration equations.
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