NATIONALEKONOMISKA INSTITUTIONEN Examensarbete C
Författare: Johannes Hagen Handledare: Henry Ohlsson Termin och år: Vårterminen 2010
On the Predictive Power of Layoffs and Vacancies
Can Advanced Notices of Dismissal and Vacancies Help Predict Unemployment?
A Study of the Swedish Labor Market Between 1988 and 2010
A
BSTRACTThe purpose of this paper is to investigate the predictive power of the variables advanced notice of dismissal (layoffs) and vacancies for the unemployment rate. Based on the Box Jenkins Methodology, the paper makes use of Granger causality and out-of-sample tests to compare the forecast performance of a naïve reference model and the two models extended to include either lagged values of layoffs or vacancies. It is shown that layoffs make up a significant leading variable, exhibiting particularly strong predictive power at forecast horizons of 2-6 months. It is also shown that the predictive power of vacancies is more ambiguous. Vacancies constitute a valuable explanatory variable for the unemployment rate, but does not possess the same leading, predictive qualities as layoffs.
Key words: ARIMA; Granger causality; out-of-sample forecasting; predictive power
C
ONTENTS1. INTRODUCTION .4
2. THEORETICAL BACKGROUND ..7
2.1CONTEMPORARY LEGISLATION .7
2.1.1LAYOFFS .7
2.1.2VACANCIES .8
2.2BOX-JENKINS METHODOLOGY .9
3. DATA .10
3.1DATA SELECTION .10
3.2CROSS-CORRELATION ANALYSIS .11
3.3STATIONARITY .12
4. MODEL SPECIFICATION ...15
4.1MODEL IDENTIFICATION ...16
4.2MODEL ESTIMATION ...17
4.3MODEL VALIDATION ...18
5. THE PREDICTIVE POWER OF LAYOFFS AND VACANCIES ...21
5.1GRANGER CAUSALITY ….22
5.2OUT-OF-SAMPLE FORECASTING ...25
5.2.1SPECIFICATION OF A FORECAST MODEL …..26
5.2.2FORECAST EVALUATION MEASURES ….28
5.2.3FIT-REGRESSION FORECASTS ...29
5.2.4SEQUENTIAL UPDATING ...33
6.ANALYSIS ...35
6.1EVALUATION OF THE PREDICTIVE POWER OF VACANCIES ...38
6.2EVALUATION OF THE PREDICTIVE POWER OF LAYOFFS ….38
6.3MODEL WEAKNESSES ...39
7.CONCLUSION ...41
8.REFERENCES ...43
APPENDIX A GRAPHS……….………..47
APPENDIX B ACF’S AND PCF’S ….49
APPENDIX C STATISTICAL TESTS ….51
APPENDIX D OUT-OF-SAMPLE FORECAST EVALUATIONS ….55
1. INTRODUCTION
It does not require too much of thought to recognize that the number of advanced notices of dismissal is positively correlated with the unemployment rate. The more people receiving advanced notice of dismissal, the greater the unemployment rate. Equally easily concluded is the fact that a high rate of reported vacancies will reflect a lower rate of unemployment. That is, the number of vacancies and the unemployment rate should exhibit a negative correlation.
Although the raw correlations are clear, there is more complexity to the relationship between the unemployment rate and these two variables. Previous research has been focused on microeconomic as well as macroeconomic aspects of these relationships. As for advanced notice, the microeconomic perspective mainly investigates whether the length of the regulated notification period affects re-employment earnings (Nord, Ting 1993, Addison, Portugal 1992) and re-employment chances (Addison, Pedro 1992, Love, Torrence 2001). The findings differ considerably, as the estimated notification effects are highly dependent on model specification and methodology (Addisson. Douglas 1992). The macroeconomic perspective, greatly influenced by the pioneering work of Edward P Lazear (Lazear 1990), is concerned with the aggregate effects on the unemployment rate of advanced notice requirements. Lazear concludes that longer notice requirements and comprehensive severance payments reduce employment-population ratios and labor force participation rates, a view later challenged by several researchers (Addison, Grosso1996).
The macroeconomic perspective has dominated the literature on vacancies. The prime focus is the Beveridge Curve, defined as the unemployment-vacancy locus (Blanchard, Diamond 1989, Pissarides 1990). The search-and-matching model of Mortensen-Pissarides has become the standard theory of unemployment equilibrium (Mortensen, Pissarides 1994), based on which empirical research about the actual movements of the Beveridge Curve has been made (Calmfors 1993, Nickell, Nunziata, Ochel, Quintini 2001). Thus, the use of vacancy data in labor economic research has primarily been motivated to shed light on the efficiency of the labor market.
Another essential question is to what extent the future labor market can be foreseen, and whether the obviously multifaceted relationships between the unemployment rate, advanced notice of dismissal and vacancies can help do this. The focus would then shift away from the economic implications of notice requirement regulation and labor market efficiency, to the
predictive and explanatory nature of advanced notice of dismissal and vacancies. In other words, are present and past data of advanced notices of dismissal and vacancies useful in forecasting the unemployment rate? Assessing the predictability of one time series variable, given information about other variables is a frequent problem in empirical work and time series modeling, but has not yet been applied to these variables.
Based on monthly data of unemployment, advanced notices of dismissal and vacancies in Sweden from 1988 to 2010, the aim of this paper is to investigate whether the variables advanced notices of dismissal and vacancies have predictive power for the unemployment rate. From now on I will refer to advanced notices of dismissals simply as layoffs.
The predictive power of an explanatory variable is the ability to improve forecasts of the dependent variable and is closely linked to the concept of causality. Causality refers “to the extent to which a processx is leading another processt y ” (Lemmens, Croux, Dekimbe 2008). t Hence, an analysis of causality and predictability as for layoffs and vacancies in relation to the unemployment rate will provide an answer to the above specified research question.
Firstly, if the number of reported layoffs or vacancies has a significant impact on the unemployment rate, one would speak of causality between these processes. However, the concept of causality, as developed by Clive Granger in 1969 (Granger 1969), does not restrict itself to immediate effects, but can also be used for more refined analysis of the exact lead- and-lag structure of time series processes. Such analysis will answer not only the question whether or not layoffs and vacancies help predict the unemployment rate, but also specifically at what time horizon this works particularly well. Testing Granger causality between two variables is based on the comparison of a reference model and an extended model. The reference model shows how much of the current unemployment rate can be explained by past values of itself. The model is then extended to include current and/or lagged values of layoffs or vacancies. If the explanation of the current unemployment rate is improved, the added independent variable is said to Granger cause unemployment, and hence exhibit predictive power. Such tests are generally referred to as in-sample test procedures, as simple t-tests or F- tests are used to determine coefficient significance.
Secondly, out-of-sample forecasting techniques shed further light on the predictive power of explanatory variables. The idea is to generate competing forecasts to compare forecast
accuracy of the reference model and the extended models. If, say, layoffs truly have predictive power for the unemployment rate, a forecast model including layoffs should be superior the reference model. Ashley, Granger Schmalensee (1979) advocate the use of out- of-sample forecast comparisons to determine the predictive power of a variable, which in practice has been common since the influential work of Meese and Rogoff (1983,1988).
Further insight into the specific time structure is provided by a cross-correlation analysis. The magnitude of the correlation coefficients between unemployment and lagged values of layoffs and vacancies indicates at what time in the future the current amount of layoffs and vacancies will be reflected in the unemployment rate.
It is important to note that the purpose of this paper is not to develop an optimal forecast model. This paper employs cross-correlation analysis, in-sample and out-of-sample test procedures in order to get a reliable and thorough picture of the predictive power of the variables layoffs and vacancies. Based on the Box-Jenkins time series methodology, I will specify an appropriate reference model that can be easily extended to include other variables.
The results show that layoffs exhibit true predictive power for the unemployment rate. Adding layoffs to the reference model particularly improves forecast accuracy at time horizons between 2-6 months, but the in-sample analysis suggests that layoffs can be effectively used even for longer forecasts. The predictive power of vacancies is more ambiguous. Although strongly negatively correlated with unemployment, vacancies exhibit no specific lead-and- lag-structure. Moreover, the Granger Causality tests reveal that vacancies barely help predict unemployment. Based on the out-of-sample forecast results where the extended vacancy model actually outperforms the reference model, vacancies are more likely to hold general, lead-and-lag independent, explanatory power for the unemployment rate. However, major vacancy data collection errors and model instability raise concern about the validity of the results.
The remainder of the paper is organized as follows. In section two a theoretical background is presented. I will present contemporary Swedish legislation regulating layoff and vacancy procedures and explain the basic ideas of the Box-Jenkins methodology. The third section describes the available data and discusses necessary transformations to make sure that the stationarity condition is fulfilled. In section four, a reference model is developed according to the model specification process as recommended by Box and Jenkins. Section five is then
devoted to test the predictive power of layoffs and vacancies for the unemployment rate, using in-sample and out-of-sample test procedures. Section six summarizes and evaluates the predictive power of layoffs and vacancies as shown by the different tests. Finally, I conclude in section seven.
2. THEORETICAL BACKGROUND 2.1 CONTEMPORARY LEGISLATION
2.1.1 LAYOFFS
The purpose of regulating advance notice of dismissal is to provide the Swedish Public Employment Service with enough time and information to facilitate labor adjustments (AMS Faktablad Varsel 2009). Advanced notice is also considered a job protection right, as the period between receiving notice and the actual dismissal increases chances of instant re- employment. Employers planning to curtail operations, which will affect more than five employees, must submit a written report to the Public Employment Service before the actual cut-down. The length of the advance notice requirement depends on the number of layoffs.
The employer must notify the Public Employment Service:
- at least two months in advance if less than 25 employees receive notice
- at least four months in advance if 26-100 people receive notice - at least six months in advance if more than 100 people receive notice Consequently, only large cut-downs are reported well in advance, whereas minor dismissals are not reported at all. After the first notification and at least one month before the definite dismissal, all downsizing employers must submit specific information about the layoffs. This allows the Public Employment Service to evaluate the number of people actually entering unemployment, which in most cases is far from the number of people originally receiving notice. In case no additional reports are sent in, the employer is assumed to have cancelled the cut-down, and kept all of his employees.
The execution rate, that is the share of notified employees eventually losing their job, tends to increase during recessions and decrease in boom times. In 2005, the rate was only 52%, which had increased to 62% in 2008, the peak of the financial crisis. However, some of these people may find another job quickly enough not to be registered as unemployed. Thus, the share of the total number of layoffs resulting in unemployment is considerably lower, ranging between 15% and 30% during the last five years (Liljegren, Jans 2010).
These numbers are very important to this paper for one reason. If a considerable share of the number of advanced notifications actually results in unemployment, the variable layoffs is expected to have a significant impact on the unemployment rate, and thus exhibit predictive power.
2.1.2 VACANCIES
The second variable under consideration is vacancies. Job vacancies are defined as “the number of unfilled job openings whose labor requirements and wage rates are formally specified in firms that are actively recruiting”(Holt, David 1966). The number of job vacancies reflects the demand of labor in an economy, or the willingness of firms to hire new workers (Pissarides 1986). Basically, an increase in the number of vacancies can be interpreted in two ways. Firstly, the demand of labor has risen, which will put more people into employment. Secondly, the search-and-match ability of the labor market has become worse, so that employers experience difficulties in finding the right personnel (Arbetsmarknadsrapport 2009:2).
To fully capture the negative correlation between vacancies and the unemployment rate, as suggested by the first interpretation, it is necessary to analyze recently reported vacancies. An increasing inflow of vacancies would trigger a fall in the unemployment rate, which is why vacancies are likely to have a lead over unemployment. In fact, it has been shown (Mortensen, Pissarides 1994) that sole anticipation of cyclical change reduces the cyclicality of job creation, that is, fewer vacancies are reported and more people enter unemployment. It is therefore reasonable to expect that recently reported vacancies hold predictive power for the unemployment rate, provided that the available vacancy data is comprehensive and reliable.
A look into contemporary Swedish legislation and the role of the Public Employment Service will shed some light on the characteristics of the vacancy data. Until a few years ago, there was a law saying that all vacancies had to be reported to the Public Employment Service.
However, the law was weakly implemented and employers could in reality choose whether or not to report vacancies (Arbetsmarknadsrapport 2009:2). Quite often, employers prefer newspapers, homepages or spontaneous applications to the Public Employment Service. Not surprisingly, the law was eventually abolished in 2007, which in accordance with what was expected, had no apparent effect on the number of reported job vacancies (Prognos Arbetsmarknad Stockholm 2007).
2.2 BOX-JENKINS METHODOLOGY
In order to evaluate the predictive power of layoffs and vacancies, we need a time series model that is appropriate for forecasting purposes. There exist many methods and approaches for formulating forecasting models (see Andersson, Joner, Ågren 2007). One of the most popular methodologies is The Autoregressive Integrated Moving Average (ARIMA) methodology as described by George Box and Gwilym Jenkins in their 1970 book entitled Time Series Analysis: Forecasting and control.
Box-Jenkins’ ARIMA models consist of two parts in order to capture the behavior of a time series: the autoregressive (AR) part and the moving average (MA) part. The autoregressive part attempts to explain the present value of the time series by means of past values of the same series, whereas the moving average part controls for past errors. Combining autoregressive and moving average components yield so called ARMA processes. Thus, a time seriesy can be modeled as a combination of pastt y values and/or past t u errors t (Makridakis 1997). The general ARMA(p,q) process is written as:
q t q t
t p t p t
t
t y y y u u u
y =α1 −1+α2 −2 +...+α − −β0 −β1 −1−...−β −
where p is the number autoregressive components and q the number of moving average components.
The theory behind ARMA processes was already known when Box and Jenkins published their book (Yule 1926, Wold 1938). However, Box and Jenkins managed to popularize the application of ARMA theory on real time series. They developed necessary computer programs and provided guidelines on how to determine an appropriate number of lags p and q, and how to optimize the parameters in the equation (Makridakis 1997). The letter ‘I’
between AR and MA stood for the ‘Integrated’ and reflected the need for differencing to make the time series stationary (Box-Jenkins 1994). The ARIMA(p,d,q) model is thus very general as it includes autoregressive models, moving average models, mixed autoregressive- moving average models, and the integrated forms of all three (Box Jenkins 1994).
The most appealing feature of the Box-Jenkins approach is that ARIMA models allow the researcher to generate accurate forecasts of a time series using only past values of this series.
If other time series data are available, these can also be included in the model to explain the behavior of the original time series (Vandaele 1983). Thus, applying the Box-Jenkins
methodology to this paper allows me to explain the present and future unemployment rate by past values of itself as well as layoff and vacancy data.
In section 3.3, I explain the concept of stationarity more in depth and investigate whether the three time series in question fulfill this condition. Section 4 then applies the Box-Jenkins methodology to develop an appropriate ARIMA model for the unemployment rate.
3. DATA
3.1 DATA SELECTION
Investigating the predictive power of advanced notices of dismissal and vacancies requires a large set of observations. The sample range for the data of all three variables is January 1988 to January 2010, totaling 265 observations.
The official unemployment rate data is published monthly by the National Bureau of Statistics (SCB) and is based on a comprehensive labor force survey (Statistics Sweden Homepage).
The data is reported as the number of thousand people registered as unemployed. Quite recently, there has been a major change to the definition of being unemployed. In accordance with ILO standards, full-time students looking for a job are now regarded as unemployed, implying a sudden increase in the unemployment rate at the date of the extension of the definition. (Konjunkturinstitutet, Fördjupning Produktion och Arbetsmarknad) In my data, this date is January, 2001. There is a clear seasonal pattern in the unemployment rate data, adjusted for by the Additive Census X12 method for monthly series. (Appendix A, Figure A.1)
The monthly layoff and vacancy data has been retrieved from the Swedish Public Employment Service (AMS Homepage). Based on a graphical analysis, there is no seasonal tendency in the layoff data, which also seems plausible from a theoretical viewpoint.
However, the amount of vacancies exhibits a seasonal pattern, giving reason to adopt the same seasonal adjustment method as above. Both data series are graphically presented in Appendix A (Figure A.2, A.3). The vacancy data includes only vacancies reported to the Public Employment Service less than 10 days back in time, since the total number of registered vacancies would reflect the efficiency of the labor market rather than the present economic situation.
3.2 CROSS-CORRELATION ANALYSIS
The relationship between layoffs and unemployment on one hand, and vacancies and unemployment on the other should be quite clear from the beginning. There is likely a positive correlation between the amount of layoffs and unemployment, particularly strong at a few months lag. As for vacancies, we can expect a negative correlation, as the number of available jobs should reflect the present unemployment rate. However, it is harder to tell at which lag the correlation is strongest. Moreover, we cannot assume that the correlation implies causality. Before going deeper into Granger causality tests and forecast evaluations to examine the exact relationships, it is useful to calculate cross-correlations of vacancies and layoffs with respect to the unemployment rate. For each variable, I compute three leads and twelve lags with respect to the unemployment rate in period t.
Table 1. Cross-correlations between unemployment, layoffs and vacancies
Lag Layoffs Vacancies
3 0.0596 -0.5515
2 0.0147 -0.5204
1 -0.0234 -0.4952
0 0.0976 -0.5766
-1 0.1393 -0.5887 -2 0.1719 -0.5972 -3 0.2054 -0.6055 -4 0.2403 -0.6091 -5 0.2654 -0.6071 -6 0.2882 -0.6060 -7 0.3139 -0.6031 -8 0.3236 -0.5965 -9 0.3302 -0.5859 -10 0.3379 -0.5713 -11 0.3330 -0.5580 -12 0.3270 -0.5398
Note: Bold numbers indicate maximum correlation coefficients for each variable
It becomes obvious that layoff data has a lead over unemployment, as the correlation increases when the layoff data is lagged. The strongest correlation can be found at lag 10. In other words, the number of layoffs today is strongest correlated with the unemployment rate in 10 months time. However, the lag pattern is not unambiguous, as previous and subsequent lags are also quite strong. Moreover, it is very important to remember that strong cross
correlations do not necessarily imply causality, and we cannot draw any conclusions about the forecast ability of layoffs. If the time series suffers from autocorrelation, the cross-correlation analysis only detects a spurious relationship. Assuming there is no autocorrelation, one can make use of the standard error to evaluate whether the correlation coefficients are significantly different from zero. The standard error is computed as 1/ n and is equal to 0,06143, where n=265. This implies that the three leads are insignificant as well as the present value of layoffs
(
α =10%)
, whereas all lags seem to correlate negatively with the unemployment rate. Thus, the cross correlation scheme confirms the belief that layoffs have a lead over unemployment.As for vacancy data, the results are more ambiguous, since all leads and lags exhibit similarly strong negative correlations. It is not clear whether vacancies have a distinct lead or lag over unemployment. Thus, rather than being able to predict the unemployment rate at a specific date in the future, the number of vacancies seem to reflect the general mood in the economy, and might therefore not be as useful as layoff data in predicting unemployment. However, the magnitude of the correlations suggests that vacancy data could Granger cause unemployment, and also be a better predictive variable than layoff data, which motivates further analysis.
3.3 STATIONARITY
Just as in regression analysis based on cross-sectional data, the major objective of time series analysis is to obtain a concise description of the features of the sample population in a mathematical model, by means of which we can draw inferences about the whole population.
However, in a time series as opposed to a cross-sectional data set, the characteristics of the population under investigation are likely to change over time, the most obvious example being the mean altered by the presence of a trend. To understand the importance of stationarity, a distinction has to be made between a time series realization and the time series process. (Vandaele 1983) The observed time series, that is the collected sample, is a realization of an underlying time series process, and in order to draw inferences about the real time series process or generate forecasts, we have to assume that the fundamental characteristics of this time series are constant through time. Box and Jenkins use the term
“statistical equilibrium” to illustrate the essence of stationarity (Box-Jenkins 1994)
If this is not true, the realization will be for a particular episode, and cannot be used to generalize about the time series process as a whole. (Gujarati 2009)
In other words, to be able to investigate the ability of layoff and vacancy data in predicting unemployment, we have to make sure that the mean and variance and autocovariance of these time series do not change over time; that is, they are stationary. Thus, there are three requirements for weak stationarity1, mathematically represented in the following manner:
Mean stationarity: E(Yt)=μ
Variance stationarity: Var(Yt)= E(Yt −μ)2 =σ2
Covariance stationarity: γk = E Y
[ (
t−μ) (
Yt+k −μ) ]
= E Y[ (
t+m−μ) (
Yt+m +k−μ) ]
If the origin of Y is shifted from Ytto Yt+m, the mean, variance and autocovariances of Yt+m must be the same as those of Yt (Gujarati 2009). The use of the Box-Jenkins methodology in modeling unemployment requires all data series to fulfill these requirements, which can be tested in three ways (Gujarati 2009). I will start with the unemployment rate.
The first and easiest test of stationarity is graphical. Plotting the number of thousand people being unemployed against time, it is easy to conclude that the mean increases with the time (Appendix A, Figure A.1). Although the unemployment decreases considerably twice during the time period 1988-2010, the recession phases seem to outdo the booms. The unemployment seems unable to return to pre-1990 levels, giving some support to the hysteresis hypotheses.
Assuming now that the mean is actually constant, it is slightly harder to tell from the graph whether the degree of variation around the mean is constant. It could be the case that the unemployment merely has drifted away and will return to its real mean level in the future (mean reversion). In this case, the unemployment exhibits mean stationarity as well as variance stationarity. An effective way to stabilize the variance is to apply the logarithmic transformation on the time series (Vandaele 1983). Plotting the transformed data reveals that the originally curved plot is now straighter (Appendix A, Figure A.4). Since the sample range is set to 22 years, we have to find other more sophisticated methods to determine the nature of the time series.
The second test is based on the autocorrelation function, visualized by a correlogram.
Autocorrelation embodies an essential feature of a time series, namely, that adjacent observations are dependent. (Box-Jenkins 1994) It is defined as the “similarity between observations as a function of the time separation between them” (Wikipedia). The
1
autocorrelation function at lag k is written as a function of the covariance at lag k and the variance: ρk =γk
γ0
= E Y
[ (
t−μ) (
Yt+k−μ) ]
E Y
[ ( t−μ)
2]
The pattern of the autocorrelations in the correlogram of the unemployment level (Appendix B, Figure B.1) is typical that of a nonstationary time series. The autocorrelation function of lag one is close to one, meaning that the unemployment in period t is highly dependent on the unemployment in period t-1. The slow decline indicates that the correlation eventually dies out, but there is clear evidence about the nonstationary nature of the unemployment. The correlogram of a stationary time series would resemble that of a white noise process (Gujarati 2009).
The third and most formal test of stationarity is the widely used Unit Root Test. In the stochastic process Yt =ρYt−1+ ut, if ρ=1, also called a unit root, the model becomes a nonstationary random walk. Accounting for the facts that the associated t-test is severely biased (Gujarati 2009) and that the error term ut is likely to be correlated, Dickey Fuller developed another test, known as the Augmented Dickey-Fuller (ADF) test. Subtracting Yt−1from both sides and adding lagged difference terms on the right hand side enables us to obtain an unbiased estimate of the unit root coefficient, δ:
ΔYt =β1+δYt−1+ αiΔYt−i+εt i=1
∑
mRejecting the null hypothesis that δ= 0, i.e., there is a unit root, implies that the time series is stationary. The appropriate number of differenced lags on the right hand side of the equation could either be based on a specific selection criterion2, or just by starting with a large number of lags and then reduce them until the last lag is significant.
The results of the ADF regression on the unemployment data in levels are as follows:
Δ lnunt = 0,07811− 0,01276lnunt−1− 0,26228Δ lnunt−1
The t-statistic of the unit root coefficient δ is equal to -1,8777, and corresponds to a p-value of 0,3426. Thus, the null hypothesis that there is a unit root cannot be rejected and we conclude that the unemployment rate is a nonstationary time series.
2The most commonly used are the Akaike and the Schwarz selection criteria
The stationarity test results naturally depend on the time interval under consideration.
Following the economic downturn in the beginning of the 1990s, there was a sharp rise in the unemployment rate, which does not seem to revert to pre-crisis levels. If the stationarity tests are conducted on a time interval excluding this regime shift, the results would probably be quite different. What can be said surely about the unemployment rate is that it is bounded.
This means that there seems to be an upper and a lower level, between which the unemployment rate fluctuates.
The three tests just described should also be applied on the other two time series, namely layoffs and vacancies. The autocorrelation functions (Appendix B, Figure B.2, B.3) and the ADF-tests (Appendix C, Test C.1, C.2) confirm that neither of them is nonstationary in levels.
Consequently, the unemployment data needs to be made stationary before being used, whereas the layoff and vacancy data can remain as they are.
There are different ways to make a data series stationary, of which Box and Jenkins advocate a method called differencing. Series that exhibit nonstationary behavior in levels may actually be similar when differences in levels are allowed (Box Jenkins 1994). Taking the first difference, that is the difference between the values of two adjacent observations, is often sufficient to obtain stationarity. According to the results of the three tests just described when applied to the differenced series, the unemployment data proves to be integrated of order one (Appendix C, Test C.3). From now on, all ARMA(p,q) models including unemployment data will be written as ARIMA(p,1,q) models, where one is the integration order.
Using models including both differenced series (unemployment) as well as such that have not been differenced (layoffs and vacancies) poses no problem when it comes to causality tests.
However, making forecasts with differenced data is not ideal, but as we shall see later, this problem can be overcome quite easily by rewriting the ARIMA(p,d,q) model to an ARMA model of a higher autoregressive order.
4. MODEL SPECIFICATION
Based on the Box-Jenkins methodology, the purpose of this section is to develop an appropriate model for testing Granger causality between layoffs/vacancies and unemployment. The model will also be used to test whether these two variables improve forecast ability.
Provided that the time series in question is stationary, Box and Jenkins encourage the use of a three-stage iterative procedure to find the most appropriate model, that is, to determine the optimal number of lags p and q, as well as the integration order of the series. The three stages are identification, estimation and validation. I will follow these steps to generate the optimal model for the unemployment rate in Sweden between 1988 and 2010. I do not take the variables layoffs and vacancies into account, as these can be added to the once fitted model to form a so called multivariate autoregressive integrated moving average specification (MARIMA) (Bagshaw 1987). The general ARIMA(p,d,q) model, from which an optimal model will be drawn, is therefore written as:
zt =α1zt−1+α2zt−2+...+αpzt−p −β0ut−β1ut−1−...−βqut−q where ztis the first-order differenced series of the transformed unemployment rate. Or more specifically:
q t q t
t p t p
t t
t un un un u u u
un = Δ − + Δ − + + Δ − − − − − − −
Δln α1 ln 1 α2 ln 2 ... α ln β0 β1 1 ... β
4.1 MODEL IDENTIFICATION
Box and Jenkins stress the uncertain nature of model identification. The identification process is “inexact because the question of what types of models occur in practice…,is a property of the behavior of the physical world and therefore cannot be decided by purely mathematical argument” (Box, Jenkins 1994). The statistician relies exclusively on graphical methods, which necessarily includes a great deal of personal judgment (Gujarati 2009).
The chief tools in identification are the autocorrelation function (ACF) and the partial autocorrelation function (PACF). The concept of autocorrelation has already been explained, whereas the partial autocorrelation is the correlation between Yt and Yt−kafter removing the effect of the intermediate Y’s. (Gujarati 2009) The ACF and the PACF for the differenced unemployment data will serve to determine the optimal number of lags p and q. If the ACF and PACF patterns are not very clear, it is difficult to choose a model without trial and error, which implies testing a selected number of ARIMA processes. (Gujarati 2009) At this stage, testing merely means checking for serial autocorrelation, and it should here be noted that identification and estimation inevitably overlap. The model that exhibits least serial autocorrelation will be exposed to more sophisticated validation tests at the last stage.
In case we deal with a pure autoregressive process of order p, its autocorrelation would tail off gradually, whereas its partial autocorrelation function would cut off after lag p. The ACF and PACF of a pure moving average process would have exactly the opposite properties (Box, Jenkins 1994). The ACF and the PACF of the differenced unemployment data reveal no clear pattern (Appendix B, Figure B.4), since the ACF as well as the PACF tail off with no distinct cutoff. This signifies that we ought to choose a mixed process to be able to turn the differenced unemployment data into an unpredictable random noise process. However, a pure AR(p) process or a MA(q) process is generally preferred to a mixed ARMA process. The main reason is the difficulties arising from parameter redundancy, or in other words, model complexity (Box Jenkins 1994). Thus, in order not to forego parsimony in parameterization, I will apply two model selection criteria on a few different ARMA processes that have proven not to exhibit any significant serial autocorrelation.
The two selection criteria are called Akaike’s Information Criteria (AIC) and Schwarz’s Information Criteria (SIC) (Appendix C, Test C4). Both criteria impose a penalty for adding regressors to the model, which is why the model with the lowest values is preferred. (Gujarati 2009)
Table 2. Akaike and Schwarz Selection Criteria for different ARIMA processes
AR(2) AR(3) ARIMA(1,1,1) ARIMA(2,1,1) ARIMA(2,1,2) Akaike -2,8253 -2.8261 -2.8174 -2.8733 -2.8658
Schwarz -2.7980 -2.7852 -2.7902 -2.8325 -2.8113 Note: The bold numbers indicate the minimum value for each selection criterion
The results confirm that a mixed process with two autoregressive lags as well as one moving average component should be chosen. Thus, based on the ACF and PACF analysis of the differenced unemployment data, and the two selection criteria, the reference model that will be used throughout the paper is an ARIMA(2,1,1) model, written as:
1 1 0 2 2 1
1 − + − − − −
= t t t t
t z z u u
z α α β β
4.2 MODEL ESTIMATION
The next step is to estimate the parameters of the chosen ARIMA model. The two available estimation methods are least squares and maximum likelihood, of which Eviews applies the non-linear version of the former. The idea of least squares estimation is to choose those values of the parameters, which will minimize the sum of the squared residual, SSR (Vandaele 1983):
( )
∑
∑
= − − −=
+
−
−
= n
t
t t
t t n
t
t z z z u
û
1
2 1 1 2 2 1 1 1
2 , , , 1 2 1
1
minα β α α β
α γ
The estimation of the identified ARIMA(2,1,1) model is presented below. The dependent variable is the transformed unemployment in first differences.
ARIMA(2,1,1) model: zt =α1zt−1+α2zt−2−β0ut −β1ut−1
Estimation: zˆt =0,588387×zt−1 +0,345338×zt−2 −0,856163×ut−1
(0,0000) (0,0000) (0,0000)
The parenthesized values below the coefficients are the corresponding p-values. Despite the complexity of the model, including three ARMA components, all coefficients are significant.
The AR coefficients are both positive, indicating that previous changes in unemployment have a positive impact on the present change in unemployment. The MA(1) term is negative, which means that the error shock in period t-1 has a negative effect on the present change in unemployment.
4.3 MODEL VALIDATION
To determine whether the chosen ARIMA(2,1,1) model is adequate for our purposes, further, more sophisticated diagnostic tests must be applied. If the tests fail to show serious discrepancies, we can feel comfortable about using the model. However, if they do indicate inadequacy, it will be necessary to restart the whole model specification process and find out in what way the model should be modified (Box & Jenkins 1994). Basically, there are three ways to test the validity of the model, namely a coefficient analysis, a stationarity analysis and a residual analysis.
Firstly, as mentioned above, parameter redundancy, or overfitting, is potentially harmful to the model. Likewise, underfitting by omitting relevant variables, would also require model modification to represent the underlying time series more efficiently. By fitting other models, either more elaborate or simpler, the chosen model is “placed in jeopardy” as Box and Jenkins (1994) put it. The easiest way to test this is to make sure all parameters are significant. As already noted, this is the case, signifying that no parameter is superfluous. When the ARIMA(2,1,1) model is extended to include more autoregressive or moving average components, these parameters become insignificant. Moreover, reducing the number of parameters causes an increase in serial autocorrelation, which suggests that the ARIMA(2,1,1) model is well specified and better than all other ARMA processes in representing the unemployment time series.
Secondly, not only the time series in question must be stationary, but also the model itself.
Computing the impulse response function to a one-time shock in the error term can test this. If the model is stationary, the effect of the shock will die out quickly, whereas the cumulated impulse responses will asymptote to its long-run value (Eviews User’s guide). Fortunately, our ARIMA(2,1,1) model proves to be stationary, after tracing a 25-period impulse response function (Appendix C, Figure C.5). Moreover, as we know, stationarity includes mean, variance as well as covariance stationarity. The two former kinds of stationarity can be tested quite easily, especially by means of graphical methods, whereas covariance stationarity might not be as intuitive. Similar to the test of a unit root in the random walk model, Yt =ρYt−1+ ut, that is ifρ=1, we can test all three roots in the ARIMA(2,1,1) model (Eviews User’s Guide).
If the model is stationary, the so-called inverse AR roots and the MA root should lie inside the unit circle. In case the absolute value of the AR roots exceed one, the model is said to be explosive, whereas the model becomes noninvertible and unusable if the MA root lies outside the unit circle. The results from the unit root test, verify the previous results of the impulse response test (Appendix C, Figure C.6).
The third aspect of model validation, and by far the most important one, is the analysis of the residuals. Since the purpose of ARIMA modeling is to filter all characteristics of a time series, we have to make sure that the only remaining, unexplained part is completely random. This part consists of the errors, defined as ut = yt−α1yt−1for an AR(1) model. The error series should be white noise, which implies that they are normally and independently distributed with mean zero, constant variance and uncorrelated over time (Vandaele 1983), summarized as: ut∼N 0,
( )
σu2γk= E u
(
tut+k)
=⎧ ⎨ σ0u2⎩
k= 0 k≠ 0
The mean assumption can easily be tested by a t-test under the null hypothesis that the mean of the residuals is equal to zero, which proves to hold (Appendix C, Test C.7). Whether the variance of the residuals is constant or not, becomes clear when plotting the residuals against time (Appendix A, Figure A.5). Obviously, there are a few outliers that seem to cause the violation of this assumption. The greatest outlier can be found in January 2001, which is a result of the change in the definition of unemployment. Normally, such technical changes would have to be controlled for by the inclusion of a dummy variable. However, since the unemployment rate is made stationary by taking first differences, such dummy variable would
be superfluous. Nonetheless, the presence of heteroscedasticity distorts coefficient estimation, and eventually harms forecast ability and precision. Outliers are difficult to handle and several solutions have been proposed (see Elsebach 1994), but extend outside the scope of my investigation.
More important is to make sure that the residuals are uncorrelated over time; there must be no serial autocorrelation. The most widely used methods to test for autocorrelation in the residuals include a graphical analysis of the autocorrelation function and partial autocorrelation function, and the more formal Breusch-Godfrey test. Firstly, if the residuals exhibit true white noise behavior, then their ACF should have no spikes (Vandaele 1983). The residual autocorrelations of the ARIMA model are all within the approximate 95% confidence limit, and therefore not significantly different from zero (Appendix B, Figure B.5) Moreover, none of the partial residual autocorrelations lie outside the confidence interval, which further emphasizes the adequacy of the model.
The more formal Breusch-Godfrey test for general, high-order ARMA processes is a complement to the popular Ljung-Box Q-statistic, also known as the Portmanteau test (Vandaele 1983). The great advantage of the Breusch-Godfrey test is that we can distinguish specifically at which lag the autocorrelation is significant, since we regress the residuals on the model’s ARMA components as well as lagged residuals. In our case, the regression looks like:
t p t i t t
t t t
t t
t z z u u u u v
u =η1 −1+η2 −2+θ1 −1+φ1 −1+φ−2 −2+...+φ− − + ,
where the coefficient φt−i measures the autocorrelation at lag t-i. If any of these single coefficients are significant, the model suffers from autocorrelation. We are also provided with an F-statistic for the joint significance of all lagged residuals. The regression output for 15 lagged residuals (p=15) is presented in the appendix, and the results confirm the conclusions based on the ACF and PACF (Appendix C, Test C.8) None of the single coefficients is significantly different from zero and the F-statistic is very low.
Finally, the last assumption about the residuals is that they should follow a normal distribution. If the normality assumption is violated, the estimators actually remain unbiased and show minimum variance (Gujarati 2009). However, inference becomes more uncertain, since the calculation of confidence intervals and significance tests for coefficients are based on the assumption of normally distributed errors. The seriousness of this violation is
nonetheless dependent on the sample size and the degree of normality. The smaller the sample, the more critical is the assumption (Gujarati 2009) and a sample of 265 observations is regarded as quite large. Moreover, if the errors are asymptotically normally distributed, the F- and t-tests become approximately valid, which is sufficient in most cases (Vandaele 1983).
Again, there are graphical methods as well as formal tests that can be used to investigate the distribution of the errors. Firstly, a histogram of the residuals reveals a shape close to that of a normally distributed population, although quite sharp (Appendix C, Test C.9). The formal test is the commonly used Jarque-Bera test statistic, which is based on two charatecteristics of the population distribution: kurtosis and skewness. Kurtosis refers to how “peaked” the distribution is, whereas the latter refers to the asymmetry of the length of the tails (Eviews User’s Guide). If either or both of these characteristics avert from what is normal, the errors will not follow a normal distribution. As seen in the appendix (Appendix C, Test C.9) the Jarque-Bera test is highly significant (p-value=0,0000), which obviously depends on the severe sharpness of the distribution (Kurtosis ≈11).
Consequently, the question naturally arises whether the non-normal error distribution depends specifically on the identified model, or on the original time series. If the underlying unemployment time series does not follow a normal distribution itself, we cannot expect the residuals of our ARIMA model to do so! In other words, if the unemployment rate is normally distributed, the ARIMA(2,1,1) model is falsely specified and must be modified. The histogram of the unemployment rate and the corresponding Jarque-Bera test reveal a non- normally distributed population (Appendix C, test C.10). Hence, we will have to be content with the relaxed assumption that the residuals are asymptotically normally distributed, and we can be confident that the specified ARIMA(2,1,1) model is “as good as it gets”.
5. THE PREDICTIVE POWER OF LAYOFFS AND VACANCIES Equipped with an appropriate reference model, we can now turn to the main question of the investigation: Does information about vacancies and layoffs improve forecasts of the unemployment rate?
The cross correlation analysis gave a first clue of the time structure between these variables, but so far nothing can be said about predictive power. Based on the concept of causality as defined by Granger (1969),i.e., the extent to which a process x is leading another process t yt, there are two ways to test this; in-sample test procedures and out-of-sample test procedures.
The in-sample test procedure makes use of the whole set of observations to regress the parameters on the dependent variable, whereas the out-of-sample test procedure divides the set of observations into two groups in order to compare “out-of-sample” forecasts of the different models. It should be noted that there is considerable disagreement among researchers about which procedure should be preferred (Chao, Corradi, Swanson, 2000), and as we shall see, both reveal weaknesses and yield somewhat different results. It is natural to perform in- sample tests before constructing out-of-sample forecasting models, as the latter procedure is far more complex.
5.1 GRANGER CAUSALITY
The in-sample test procedure was originally developed by Granger, which is why one speaks of Granger Causality tests rather than in-sample tests. The idea is quite straightforward and can be summarized easily as follows: “xt Granger causes ytif past values of xt helps to predict yt” (Chao, Corradi, Swanson, 2000). In other words, we want to know how much of the current variable ytcan be explained by its own past values, and whether adding lagged values of x improves the explanation. The mathematical representation would look like:
Reference model:
∑
= −
= p
j
p t j
t y
y
1
α*
Extended model: yt = α*jyt− p+
j=1
∑
p β*jxt− j+ ut j=1∑
qThe coefficients of interests are the βj
*: s. The variablextdoes not Granger causeytif the βj
*: s are jointly insignificant, meaning that the null hypothesis H0:β1
*=β2
* = ...=βj
*= 0 is not rejected. Also of interest is to investigate whether the sum of the coefficients is significantly different from zero: H0 :β1*+β2* +...+β*j =0. Finally, the significance of single lags could also be of interest, which would imply the use of simple t-test statistics rather than a comprehensive Wald- or F-statistic.
In our case, the reference model is the specified ARIMA(2,1,1), to which the present and lagged values of layoffs or vacancies are added to form the extended model, now a so-called MARIMA model. For example, if we want to know whether the first three lags in layoffs are jointly significant and thereby Granger cause unemployment, the following regressions and tests are computed:
Reference model: zt =+α1zt−1+α2zt−2 −β0ut −β1ut−1
Extended model: zt =γ1Dt +α1zt−1+α2zt−2 +δ1Lt−1+δ2Lt−2 +δ3Lt−3 −β0ut −β1ut−1 0
: 1 2 3
0 δ +δ +δ =
H
HA :δ1 +δ2 +δ3 ≠0
If the three coefficients of the lagged layoff data together Granger cause unemployment, the null hypothesis must be rejected. The three first lags in layoffs prove to be jointly significant3, which is not the case for three first lags in vacancies4. Hence, we can be quite confident that layoffs have some degree of predictive power for the unemployment rate, whereas the predictive nature of vacancies needs further investigation. The Granger Causality analysis is therefore extended to include up to twelve lags of both variables. Testing the significance of the sum of a specific set of lagged coefficients will help shed light on the predictive power of layoffs and vacancies. These in-sample test results will help us define the set of lags that has most predictive power for the unemployment rate. This set of lags will later be used for out- of-sample forecasting.
Table 3. Granger Causality tests for the joint significance of lags of layoffs and vacancies
Layoffs Vacancies
Lags Coefficient F-stat P-value Coefficient F-stat P-value 0-1 2.39E-06 13.9844 0.0002 6.81E-08 0.1542 0.6948 0-2 2.27E-06 13.9222 0.0002 1.12E-07 0.5078 0.4767 0-3 2.27E-06 13.7376 0.0003 1.17E-07 0.5435 0.4617 0-4 2.30E-06 13.7721 0.0003 1.35E-07 0.8938 0.3453 0-5 2.00E-06 21.8791 0.0000 1.36E-07 1.2617 0.2624 0-6 2.10E-06 12.2302 0.0006 1.41E-07 1.6327 0.2025 0-7 2.03E-06 13.1392 0.0004 9.20E-08 1.8361 0.1767 0-8 1.92E-06 13.5479 0.0003 1.49E-07 3.2095 0.0745 0-9 1.84E-06 17.8467 0.0000 1.30E-07 3.1656 0.0765 0-10 1.89E-06 19.3052 0.0000 1.25E-07 3.392 0.0668 0-11 1.79E-06 16.6171 0.0001 1.25E-07 1.4996 0.2219 0-12 1.71E-06 17.0306 0.0001 1.30E-07 3.9736 0.0474 Note. The bold numbers indicate significant lag combinations
(
α=10%)
It is obvious that layoffs Granger cause unemployment. All F-tests are highly significant independent of lag combination. The peak of the causality pattern is reached when the present and the first five lagged values of the layoff rate are added to the reference model (p- value=0,0000). Adding further lags neither worsen nor improve the explanatory power of the model.
3F-statistic, P-value: (32,1615, 0,0000)
4
Vacancies exhibit a quite different pattern. The first lag combinations are highly insignificant and it therefore seems as if the first lags have no predictive power for the unemployment rate at all. The inclusion of lags greater than six yields slightly more significant coefficients. We can therefore assume that lags of high orders have predictive power for the unemployment rate. Further F-tests for joint significance reveal that the sum of lags 7-10 diverts most strongly from zero of all possible lag combinations5.
What we also want to know is the exact time structure between the unemployment rate and these two variables. Subsequently, instead of adding several lags of the x-variable on the right hand side of the equation, individual lags are added. In this way, the strongest lead of layoffs (vacancies) over unemployment can be found. Table 4 reports the value of the coefficient, the T-statistic and the p-value of the present value as well as 12 lags of layoffs and vacancies respectively.
Table.4 Granger Causality tests for individual lagged values of vacancies and layoffs
Layoffs Vacancies
Lags Coefficient T-statistic P-value Coefficient T-statistic P-value 0 2.24E-06 3.6966 0.0003 -7.96E-07 -2.2386 0.0261 1 2.33E-06 3.8028 0.0002 8.62E-08 0.4926 0.6226 2 2.02E-06 3.4744 0.0006 1.44E-07 0.8524 0.3947 3 1.85E-06 3.1015 0.0021 1.25E-07 0.7509 0.4534 4 1.86E-06 3.0734 0.0023 1.72E-07 1.0292 0.3043 5 1.51E-06 2.2791 0.0235 2.18E-07 1.2524 0.2116
6 1.15E-06 1.554 0.1214 2.20E-07 1.2906 0.1981
7 1.17E-06 1.6019 0.1104 2.70E-07 1.5648 0.1189
8 9.09E-07 1.1647 0.2452 3.02E-07 1.7291 0.0851
9 5.59E-07 0.618 0.5371 2.65E-07 1.5992 0.1112
10 9.01E-07 1.0448 0.2971 2.58E-07 1.3967 0.1637 11 5.31E-07 0.5464 0.5852 2.57E-07 1.4689 0.1431 12 2.88E-07 0.2785 0.7808 3.05E-07 1.7015 0.0901 Note. The bold numbers indicate which lags are significant
(
α =10%)
What conclusions can be drawn from the results?
Starting with vacancies, only the present value is significant, after which an unclear pattern follows. The eight lag, for example, is significant at a 10% significance level, whereas the first to the seventh lags are highly insignificant. Thus, the predictability improvement of unemployment is rather small when single lags of vacancies are added to the model. Although the cross correlation analysis showed that the correlation between unemployment and lagged values of vacancies is quite strong, the in-sample test procedure makes clear that correlation
5F-statistic, P-value: (6,8687, 0,0093)
does not imply causation. Given the unclear time structure in the cross correlation table (all lags are equally strongly correlated with unemployment), the coefficients estimated above should be interpreted with care. We should therefore stick to a set of lags rather than individual lags when interpreting the predictive power of vacancies.
As for layoff data, the results are easier to interpret. Lags 0-5 are all very significant, after which the magnitude of the causality diminishes. This pattern seems plausible, as most advanced notices are reported 2-4 months prior to the actual dismissal. Layoffs reported today do not directly cause a rise in the unemployment further into the future than six months.
As for the reliability of in-sample test procedures, some concern must nonetheless be raised when interpreting the results. Extra care should be taken as in-sample tests generally reject the null hypothesis of no predictability more often than out-of-sample tests. The reason is that in- sample tests are biased in favor of detecting spurious predictability (Inoue, Kilian 2002).
Similar to the concept of spurious regression, spurious predictability means identifying the explanatory value of past values of the variablextin predictingyt, when there is in effect none. This weakness is sufficient for many researchers not to rely solely on in-sample test procedures, nicely formulated by Ashley, Granger and Schmalansee (1979):
“…a sound and natural approach to such tests [Granger causality tests] must rely primarily on the out-of-sample forecasting performance.” (p.1156)
“…the riskiness of basing conclusions about causality…entirely on within-sample performance is reasonably clear” (p.1149)
If the results are not supported by out-of-sample tests, the likelihood of spurious inferences grows big (Inoue, Kilian 2002). In our case, the risk is that we might falsely accept vacancies and layoffs as valuable predictive variables.
5.2 OUT-OF-SAMPLE FORECASTING
It is intuitively clear that the predictive power of layoffs and vacancies is most efficiently evaluated by means of real forecasts. The Granger causality tests can indeed be viewed as tests of predictive ability, but evaluating and comparing actual forecast ability of different models will provide a more refined answer (Chao, Corradi, Swanson 2000). The main idea is to compare the forecast ability of our ARIMA(2,1,1) reference model, to that of the extended
