Forecasting the US unemployment rate

doi:10.1016/S0167-9473(02)00230-X

Computational Statistics & Data Analysis

Volume 42, Issue 3, 28 March 2003, Pages 451-476

https://doi.org/10.1016/S0167-9473(02)00230-X Get rights and content

Abstract

The primary interest is in out-of-sample forecasting of the US monthly unemployment rate. Several linear unobserved components models are fitted and their comparative forecasting accuracy is assessed by means of an extensive rolling-origin procedure using a test period that covers the last two decades. An attempt is made to link forecasting performance to the time domain properties of the models and the evidence is that highly persistent models perform better. Deletion diagnostics and normality tests, along with documenting possible departures from linearity and Gaussianity attributable to business cycle and turning point asymmetries, foster the conclusion that these are mostly concentrated in the pre-forecast period (1948–1980). A search is made for plausible nonlinear extensions capable of accounting for dynamic asymmetries in unemployment rates, leading to the specification of a cyclical trend model with smooth transition in the underlying parameters that improves forecast accuracy at short lead times and at the end of the sample period; as expected, though significant, the gains are not exceptionally large. The generalised impulse response function casts some light on the interpretation of the results. In particular, the main evidence is that persistence is not a stable feature over the business cycle.

Introduction

The US unemployment rate represents a case study in nonlinear dynamics. Asymmetric behaviour over the course of the business cycle has been documented in a variety of papers including Neftçi (1984), DeLong and Summers (1986) and Rothman (1991), who deal with the type of asymmetry named steepness, taking place when contractions are steeper than expansions: this implies that unemployment rates rise faster than they decrease. Sichel (1993) found evidence for deepness, which occurs when contractions are deeper than expansions, so that the amplitude of peaks in unemployment rates exceeds that of troughs; McQueen and Thorley (1993) detect turning point asymmetry (sharpness), such that peaks are sharp and troughs are more rounded.

Needless to say, the series represents a testbed for nonlinear time series models; within the class of regime switching models, threshold autoregressive models (TAR) are prominent; references include Hansen (1997), Koop and Potter (1999), who focus on the monthly unemployment rate for males aged 20 and over, and Montgomery et al. (1998). The latter conduct a rolling forecast experiment for the rates, which shows that TAR and Markov-switching models outperform the linear ARIMA benchmark model during periods of rapidly increasing unemployment, but not globally; moreover, they find that using monthly data for forecasting quarterly rates improves the forecasting accuracy only in the short term.

Skalin and Teräsvirta (1999) use a logistic smooth transition autoregressive model (LSTAR) for the first differences of the unemployment rate of OECD countries including a lagged level term and were unable to reject linearity for the US quarterly seasonally unadjusted series. Their specification assumes that the series is globally stationary, but possibly nonlinear and locally nonstationary. van Dijk et al. (2000) apply the model to the seasonally unadjusted monthly series for males aged 20 and perform an out-of-sample forecast accuracy analysis showing that the LSTAR model outperforms the linear AR counterpart at long run forecast horizons during downturns and at short run horizons during expansions.

Rothman (1998) compares the out-of-sample forecasting accuracy of six nonlinear models and finds that results are sensitive to the detrending issue. Parker and Rothman (1997) model the quarterly adjusted rate by an AR(2) process including an explanatory variable measuring the current depth of recession, and show with a rolling forecast experiment that significant reductions of the forecast MSE are achieved.

The primary interest of this paper is in out-of-sample forecasting for the US unemployment rate. The series considered is the monthly seasonally adjusted series, defined as the ratio of the seasonally adjusted unemployment level and the civilian labour force level provided by the Bureau of Labor Statistics (BLS). It is displayed in Fig. 1. Prima facie the plot confirms the presence of dynamic asymmetries in the form of steepness, which appears to be the dominant feature of the cycle dynamics during the seventies and the mid-fifties; the last two decades and some cyclical patterns at the beginning of the sample period and around 1962 seem to be more characterised by deepness and sharpness, in combination with moderate steepness, since a period of rapidly increasing rates is followed by a steep decrease. Perhaps a three phases characterisation is enforced here with the third phase representing a more prolonged and moderate decline in unemployment rates. Another feature of interest is the tendency of the series to remain on a level it has reached, with no apparent tendency to return to a stable underlying level; this is referred to as hysteresis or persistence.

Our objective is twofold. In the first place, we aim at assessing the role of persistence in forecasting within a linear framework: when the labour market is perturbed by a shock, unemployment reaches a new level; the level that will be attained (which coincides with the eventual forecast function) depends on persistence, which is a function of the model parameters; the latter also govern the speed of the transition to the new level. The evidence is that models that imply high persistence fare better from the predictive standpoint, which provides support for the hysteresis hypothesis.

Secondly, we aim at assessing the relevance of business cycle asymmetries for forecasting purposes: the main finding is that persistence is not a stable feature over the business cycle, and a nonlinear specification is capable of producing more accurate forecasts. The generalised impulse response function, proposed by Koop et al. (1996), illustrates quite effectively this finding.

Our approach is much in the same spirit of Montgomery et al. (1998) in that it focuses on an in-depth comparison of forecasting models (with an emphasis on short term forecasting), aimed at providing an understanding of the strengths and weaknesses of each. With respect to the above article we concentrate on monthly rather than quarterly data, extend the out-of-sample forecast comparison and adopt an unobserved components modelling approach.

We consider the seasonally adjusted series to enhance the comparison with other studies of the US unemployment rate, which dealt with the same series; furthermore, modelling seasonality produces a relevant increase in the computational burden of the rolling forecast experiment considered in the paper (especially in the nonlinear case).

The outline of the paper is as follows: Section 2 introduces linear unobserved components models for forecasting the US unemployment rate and illustrates their basic properties, along with the implied impulse response function. Forecasting accuracy is validated in Section 3 by a rolling forecast experiment conducted over the period 1980.1–2000.12. A comparison is also made with the ARIMA benchmark adopted by Montgomery et al. (1998). The best performance is provided by a trend plus irregular model with the trend specified as a highly persistent ARIMA(1,1,0) process.

We then document departures from linearity and normality (Section 4) using various residuals and deletion diagnostic resulting from the linear model fit. The conclusion is that these are most prominent in the test period. In Section 5 nonlinear alternatives are specified to account for dynamic asymmetries in unemployment rates. This is done by imposing smooth transition on the parameters of the model. A new transition variable that is better behaved is introduced and the results are discussed.

In Section 6 we deal with forecasting with nonlinear structural models and report the outcome of the rolling forecast experiment; these show that a nonlinear cyclical trend model outperforms the linear benchmark at short run forecast horizons. Interestingly, the improvement is greater at the end of the sample period. To interpret this finding and to gain a better understanding of the dynamic properties of the model, we find the generalised impulse response function (Section 7) quite helpful. Section 8 concludes the paper.

Section snippets

Linear structural models

This section focuses on five linear forecasting models for the levels of the unemployment rate; the models entertained assume that hysteresis arises from the presence of a unit root in the reduced form; stationarity tests and unit root tests, although the latter are not directly relevant here due to the presence of moving average features, support this assumption. The uncertainty lies in the characterisation of the persistence and in the modelling of the short run dynamics; so the approach we

Comparative performance of rolling forecasts for linear models

We use a rolling forecast experiment as an out-of-sample test of forecast accuracy. As pointed out in Tashman (2000) a most crucial issue is how to split the series between the pre-forecast and the test period. Assuming that our interest lies in short run forecasting so that the greatest lead time is 12 months, we have decided to use the sample period 1948.1–1979.12 as the pre-forecast period and to leave the last 21 years of monthly observations for evaluating and comparing the out-of-sample

Leave-k-out diagnostics and nonlinearity

Departure from normality was detected for the innovations of the ARTM; the breakdown into the contribution of the two terms skewness and kurtosis reveal that the latter is mostly responsible for the recorded high value.

Asymmetric behaviour with respect to the business cycle phases would show up in a skewed distribution for the auxiliary residuals (see Harvey and Koopman, 1992). These are estimators of the disturbances associated with the components, conditional on the entire information set,

Nonlinear cyclical trend models

In this section, we consider four nonlinear trend models, derived from ARTM and CTM, that can account for dynamic asymmetries in unemployment rates, such as those arising when unemployment rates are characterised by steep increases during recessions and slower declines during expansions. The models belong to the class of smooth transition structural time series models (see Proietti, 1999), according to which we let the fundamental parameters vary according to the state of the system, as

Forecasting with nonlinear structural models

One-step-ahead forecasts are immediately available from the KF output at the end of the pre-forecast period; multistep forecasts, conditional on the estimated parameters, are generated by the Monte Carlo method (see Granger and Teräsvirta, 1993, Chapter 8) as $y ̃_{t+l|t} = 1 M ∑ i=1 M y ̃_{t+l|t}^{(i)},$ which, for i=1,…,M, requires the following steps:

1.	draw	$y_{t+1}^{(i)} ∼f(y_{t+1} \|Y_{t})$
2.	draw	$y_{t+2}^{(i)} ∼f(y_{t+2} \|Y_{t},y_{t+1}^{(i)})$
$⋮$	$⋮$	$⋮$
$l−1$ .	draw	$y_{t+l−1}^{(i)} ∼f(y_{t+l−1} \|Y_{t},y_{t+1}^{(i)},…,y_{t+l−2}^{(i)})$
l.	evaluate	$y ̃_{t+l\|t}^{(i)} = E (y_{t+l} \|Y_{t},y_{t+1}^{(i)},…,y_{t+l−1}^{(i)})$

Generalised impulse response function

The notion of a generalised impulse response function (GIRF) provides further insight into the dynamic properties of CTMStD. The underlying idea is that in a nonlinear system the impact of an innovation occurring at time t on the future path of the process depends crucially upon the history of the process and the size of the innovation.

According to the definition by Koop et al. (1996), $GIRF (l,ν_{t},Y_{t−1})=E[y_{t+l} |ν_{t},Y_{t−1}]−E[y_{t+l} |Y_{t−1}],$ where the notation stresses the dependence on the lead time l, the

Concluding remarks

This paper has investigated the out-of-sample performance of linear and nonlinear structural time series models of the US unemployment rate by means of an extensive rolling forecast experiment using a test period made up of the last two decades.

The first conclusion is that our study corroborates the hysteresis hypothesis, as we found that linear models characterised by higher persistence perform significantly better. The best performance was provided by a trend plus irregular model with the

Acknowledgements

This research was supported by Cofin. MURST 2000—prot. MM13035581_003 as part of the project Linearity and Nonlinearity in Time Series Dynamics. The paper was presented at the Econometrics Seminars, Tinbergen Institute, Amsterdam, the School of Finance and Economics, University of Technology Sydney, the 2001 Econometric Society Australasian Meeting, University of Auckland, New Zealand, the European Central Bank, and the SCO 2001 Conference in Brixen, Italy. I would like to thank the

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View full text

Forecasting the US unemployment rate

Abstract

Introduction

Section snippets

Linear structural models

Comparative performance of rolling forecasts for linear models

Leave-k-out diagnostics and nonlinearity

Nonlinear cyclical trend models

Forecasting with nonlinear structural models

Generalised impulse response function

Concluding remarks

Acknowledgements

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