Forecasting the US unemployment rate

Computational Statistics & Data Analysis 42 (2003) 451–476www.elsevier.com/locate/csda

Forecasting the US unemployment rate

Tommaso Proietti∗

Dipartimento di Scienze Statistiche, University of Udine, Via Treppo 18, Udine 33100, Italy

Received 1 December 2001; received in revised form 1 May 2002

Abstract

The primary interest is in out-of-sample forecasting of the US monthly unemployment rate.Several linear unobserved components models are 0tted and their comparative forecasting ac-curacy is assessed by means of an extensive rolling-origin procedure using a test period thatcovers the last two decades. An attempt is made to link forecasting performance to the timedomain properties of the models and the evidence is that highly persistent models perform bet-ter. Deletion diagnostics and normality tests, along with documenting possible departures fromlinearity and Gaussianity attributable to business cycle and turning point asymmetries, foster theconclusion that these are mostly concentrated in the pre-forecast period (1948–1980). A searchis made for plausible nonlinear extensions capable of accounting for dynamic asymmetries inunemployment rates, leading to the speci0cation of a cyclical trend model with smooth transitionin the underlying parameters that improves forecast accuracy at short lead times and at the endof the sample period; as expected, though signi0cant, the gains are not exceptionally large. Thegeneralised impulse response function casts some light on the interpretation of the results. Inparticular, the main evidence is that persistence is not a stable feature over the business cycle.c© 2002 Elsevier Science B.V. All rights reserved.

Keywords: Structural time series models; Nonlinearity; Forecasting performance; Persistence; Leave-k-outdiagnostics; Generalised impulse response function

1. Introduction

The US unemployment rate represents a case study in nonlinear dynamics. Asymmet-ric behaviour over the course of the business cycle has been documented in a varietyof papers including Neft=ci (1984), DeLong and Summers (1986) and Rothman (1991),who deal with the type of asymmetry named steepness, taking place when contractions

∗ Tel.: +39-432-249-581; fax: +39-432-249-595.E-mail address: [email protected] (T. Proietti).

0167-9473/03/$ - see front matter c© 2002 Elsevier Science B.V. All rights reserved.PII: S0167 -9473(02)00230 -X

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452 T. Proietti / Computational Statistics & Data Analysis 42 (2003) 451–476

are steeper than expansions: this implies that unemployment rates rise faster than theydecrease. Sichel (1993) found evidence for deepness, which occurs when contractionsare deeper than expansions, so that the amplitude of peaks in unemployment rates ex-ceeds 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 focuson 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 thatTAR and Markov-switching models outperform the linear ARIMA benchmark modelduring periods of rapidly increasing unemployment, but not globally; moreover, they0nd that using monthly data for forecasting quarterly rates improves the forecastingaccuracy only in the short term.Skalin and TerGasvirta (1999) use a logistic smooth transition autoregressive model

(LSTAR) for the 0rst diHerences of the unemployment rate of OECD countries in-cluding a lagged level term and were unable to reject linearity for the US quarterlyseasonally unadjusted series. Their speci0cation assumes that the series is globally sta-tionary, but possibly nonlinear and locally nonstationary. van Dijk et al. (2000) applythe model to the seasonally unadjusted monthly series for males aged 20 and performan out-of-sample forecast accuracy analysis showing that the LSTAR model outper-forms the linear AR counterpart at long run forecast horizons during downturns and atshort run horizons during expansions.Rothman (1998) compares the out-of-sample forecasting accuracy of six nonlinear

models and 0nds that results are sensitive to the detrending issue. Parker and Rothman(1997) model the quarterly adjusted rate by an AR(2) process including an explanatoryvariable measuring the current depth of recession, and show with a rolling forecastexperiment that signi0cant reductions of the forecast MSE are achieved.The primary interest of this paper is in out-of-sample forecasting for the US unem-

ployment rate. The series considered is the monthly seasonally adjusted series, de0nedas the ratio of the seasonally adjusted unemployment level and the civilian labour forcelevel provided by the Bureau of Labor Statistics (BLS). It is displayed in Fig. 1. Primafacie the plot con0rms the presence of dynamic asymmetries in the form of steepness,which appears to be the dominant feature of the cycle dynamics during the seventiesand the mid-0fties; the last two decades and some cyclical patterns at the beginningof the sample period and around 1962 seem to be more characterised by deepness andsharpness, in combination with moderate steepness, since a period of rapidly increasingrates is followed by a steep decrease. Perhaps a three phases characterisation is en-forced here with the third phase representing a more prolonged and moderate decline inunemployment rates. Another feature of interest is the tendency of the series to remainon a level it has reached, with no apparent tendency to return to a stable underlyinglevel; this is referred to as hysteresis or persistence.Our objective is twofold. In the 0rst place, we aim at assessing the role of persis-

tence in forecasting within a linear framework: when the labour market is perturbedby a shock, unemployment reaches a new level; the level that will be attained (which

T. Proietti / Computational Statistics & Data Analysis 42 (2003) 451–476 453

1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000

3

4

5

6

7

8

9

10

11

Fig. 1. US unemployment rate, January 1948–December 2000; seasonally adjusted series.

coincides with the eventual forecast function) depends on persistence, which is a func-tion of the model parameters; the latter also govern the speed of the transition to thenew level. The evidence is that models that imply high persistence fare better from thepredictive standpoint, which provides support for the hysteresis hypothesis.Secondly, we aim at assessing the relevance of business cycle asymmetries for fore-

casting purposes: the main 0nding is that persistence is not a stable feature over thebusiness cycle, and a nonlinear speci0cation is capable of producing more accurateforecasts. The generalised impulse response function, proposed by Koop et al. (1996),illustrates quite eHectively this 0nding.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 shortterm forecasting), aimed at providing an understanding of the strengths and weaknessesof each. With respect to the above article we concentrate on monthly rather thanquarterly data, extend the out-of-sample forecast comparison and adopt an unobservedcomponents 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 therolling 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 ac-curacy is validated in Section 3 by a rolling forecast experiment conducted overthe period 1980.1–2000.12. A comparison is also made with the ARIMA benchmarkadopted by Montgomery et al. (1998). The best performance is provided by a trendplus irregular model with the trend speci0ed 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 0t. The conclusion isthat these are most prominent in the test period. In Section 5 nonlinear alternatives arespeci0ed to account for dynamic asymmetries in unemployment rates. This is done byimposing smooth transition on the parameters of the model. A new transition variablethat 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 trendmodel outperforms the linear benchmark at short run forecast horizons. Interestingly,the improvement is greater at the end of the sample period. To interpret this 0ndingand to gain a better understanding of the dynamic properties of the model, we 0nd thegeneralised impulse response function (Section 7) quite helpful. Section 8 concludesthe paper.

2. Linear structural models

This section focuses on 0ve linear forecasting models for the levels of the unem-ployment rate; the models entertained assume that hysteresis arises from the presenceof a unit root in the reduced form; stationarity tests and unit root tests, although thelatter are not directly relevant here due to the presence of moving average features,support this assumption. The uncertainty lies in the characterisation of the persistenceand in the modelling of the short run dynamics; so the approach we take in this sec-tion and the next is that of exploring the strengths and de0ciencies of each forecastingmodel.The models are hereby described; for more details on their time and frequency

domain properties the reader is referred to Harvey (1989).

(1) Local level model (LLM):

yt = �t + �t ; t = 1; 2; : : : ; T; �t ∼ NID(0; 2� )

�t+1 = �t + t ; t ∼ NID(0; 2); (1)

where NID denotes normally and independently distributed. The series is decom-posed into a trend component, �t , represented by a random walk with NID dis-turbances, and an irregular component, �t ; the reduced form is IMA(1,1), withnegative MA coeOcient, and thus the model implies that persistence is not greaterthat one. If 2

� = 0 the model produces na23ve or no change forecasts, whereas inthe case 2

� ¿ 0 it yields exponential smoothing forecasts (Muth, 1960).


(2) Local linear trend model (LLTM):

yt = �t + �t ; �t ∼ NID(0; 2� ); t = 1; 2; : : : ; T;

�t+1 = �t + �t + t ; t ∼ NID(0; 2);

�t+1 = �t + t ; t ∼ NID(0; 2 ); (2)

where �t is the stochastic slope, and t , t , �t are independent of one another.The trend is now given by an IMA(2,1) process, with the Hodrick and Prescott(1997) trend arising as a special case (2

=0, smoothing parameter �=2� =

2 ). The

reduced form is a restricted IMA(2,2) model and the forecasts function follows aHolt and Winters updating scheme (Holt, 1957; Winters, 1960).

(3) Trend plus cycle model (TpCM):

yt = �t + t + �t ; �t+1 = �t + t ;

the cyclical component, t , is speci0ed by the stochastic diHerence equation:[ t+1

∗t+1

]= �

[cos �c sin �c

−sin �c cos �c

][ t

∗t

]+

[�t

�∗t

]; (3)

where �t ∼ NID(0; 2�) and �∗

t ∼ NID(0; 2�) are mutually independent and inde-

pendent of t ; �t at all times; �∈ [0; 1) is the damping factor and �∈ [0; �] is thecycle frequency; t has univariate ARMA(2,1) representation such that the rootsof the AR polynomial are a pair of complex conjugates with modulus �−1 andphase �c; correspondingly, the spectral density displays a peak at �c. The modelembodies the notion that unemployment can be decomposed orthogonally into anatural rate (permanent component) and a transitory component; see Staiger et al.(1997).

(4) Cyclical trend model (CTM):

yt = �t + �t ; �t+1 = �t + t ;

where t is given in (3), so that the trend is an ARIMA(2,1,1) process. Thus,cyclical movements are integrated within the trend; the model is consistent withan alternative de0nition of business cycle Ructuations, that are not considered interms of deviations from the trend component. The reduced form is, as for TpCM,an ARIMA(2,1,2) model, but the implications are quite diHerent: TpCM impliesthat the spectral density of the 0rst diHerences is a minimum at zero, whereas thisrestriction is not enforced by the CTM, which allows the innovations to be morepersistent than a random walk. With respect to LLM, the trend disturbances areserially correlated (a stationary ARMA(2,1) process), which allows the clusteringof positive and negative disturbances during upswings and downswings.

(5) Autoregressive trend model (ARTM):

yt = �t + �t ; �t+1 = �t + t ; t+1 = � t + �t: (4)

This is the same as CTM with �c = 0; the recursion for ∗t becomes redundant

and is dropped; the model is also referred to as a damped slope trend model.


Table 1Parameter estimates and diagnostics for linear structural models of US monthly unemployment, 1948.1–2000.12

LLM LLTM TpCM CTM ARTM

2 0.0469 0.0110 0.0213

2 0.0041

2� 0.0186 0.0050 0.0085

� 0.9825 0.8723 0.7853�c 0.1159 0.18662� 0.0000 0.0093 0.0000 0.0135 0.0122

loglik 69.38 89.42 94.86 119.47 115.75Q(12) 170.74 42.25 74.30 23.03 30.18Q(24) 205.28 65.93 90.14 48.34 51.56Normality 348.63 417.24 499.91 359.57 436.07

The eventual forecast function is horizontal, except for LLTM, for which it is a straightline; persistence is relevant in that it determines the level or the slope of the eventualforecast function.The model design is

Some of the representations are nested and some are not (for instance TpCM andCTM); even for nested models (LLTM reduces to LLM when 2

= 0 and �t = � = 0)model selection and hypothesis testing constitute non standard issues and the reader isreferred to Harvey (1989, Chapter5) and Harvey 2000, and for these topics. As statedbefore, the main interest of the paper is selecting a forecasting model according to itspost-sample performance.The reference framework for statistical treatment is the linear Gaussian state space

representation and likelihood inference is carried out by means of the Kalman 0lter (seeAppendix A.1). Parameters estimates for the full sample (1948.1–2000.12), along withsome basic diagnostics and goodness of 0t statistics computed on the KF standardisedinnovations, are reported in Table 1. 1

For the LLM the irregular variance was estimated to be zero, and therefore the modelreduces to a simple random walk and provides naGSve or “no change” forecasts; in therolling forecast experiment of the next section it will thus be taken as the bench-mark against which the performance of the other models will be assessed. It suHersfrom various forms of misspeci0cation as highlighted by the Box–Ljung portmanteau

1 All the computations were performed using the library of state space functions SsfPack 2.3 by Koopmanet al. (1999), linked to the object oriented matrix programming language O x 2.1 of Doornik (1998).


statistics, Q(12); Q(24), and by the normality test recommended by Doornik and Hansen(1994) based on Bowman and Shenton (1975).The additive cycle estimated by TpCM has a period of about 4 years and a half

(54 months) and is characterised by a value of the damping factor which is close toone. CTM and ARTM produce smaller Box–Ljung statistics, which are neverthelesssigni0cant; the signi0cance is due to the negative residual autocorrelations at lags 12and 24, that are a likely consequence of overadjustment produced by X-11-ARIMA. Ina previous version of the paper, we also considered an extension of CTM with a sea-sonal feature: namely, we replaced the irregular component with a seasonal ARMA(1,1)process, but it was found that this improves only the in-sample 0t with virtually noconsequence for forecast accuracy. All models suHer from serious departure from nor-mality. 2 Further discussion is provided in Section 4.The impulse response function (IRF) is a standard tool in illustrating the dynamic

properties of the time series models we have entertained; namely, it examines the eHectof an innovation occurring at time t, �t = yt − E[yt |Yt−1], where Yt−1 = {y1; : : : ; yt−1}is the information up to and including time t − 1, on the future pattern, yt+l; l¿ 0,by looking at the sequence of dynamic multipliers:

@yt+l

@�t; l= 0; 1; : : : :

For the linear and time invariant models of this section the IRF is a function of l alone(and the model parameters), and it is readily computed from the steady-state KF (seeAppendix A.1), being provided by the sequence {1; ZK; ZTK; ZT 2K; : : : ; ZT l−1K; : : :}.

Fig. 2 shows the IRF of a selection of models; for LLTM it diverges to in0nityat a linear rate. It should be noted that for TpCM the IRF describes a damped waveconverging rather slowly to a value (persistence) below one; this is a consequence ofthe orthogonal trend-cycle decomposition. Cyclical trend models imply high persistence,with the ARTM yielding the highest pro0le. The 0gure also reports the IRF pattern forthe ARIMA model considered by Montgomery et al. (1998); this will be discussed inSection 3.

3. 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 seriesbetween the pre-forecast and the test period. Assuming that our interest lies in shortrun forecasting so that the greatest lead time is 12 months, we have decided to usethe sample period 1948.1–1979.12 as the pre-forecast period and to leave the last 21years of monthly observations for evaluating and comparing the out-of-sample fore-cast performance of the various alternative models. The test period includes two full

2 Two other models were entertained: the 0rst is the LLTM with 0xed slope, that is (2) with �t = �;the second is CTM with a 0xed slope, that is with trend component �t+1 = �t + � + t . In both case theestimated slope was not signi0cantly diHerent from zero and so the models reduce, respectively, to LLMand CTM.


0 5 10 15 20 25 30 35 40 45 50 55 60

0.50

0.75

1.00

1.25

1.50

1.75

2.00

LLM

TpCM

ARTM

CTM

ARIMA

LLM CMT ARIMA

TpCM ARTM

Fig. 2. Impulse response function for linear models of US unemployment rate (ARIMA refers to the model0tted by Montgomery et al. (1998), see Section 3).

cycles in the unemployment rate and the long lasting expansion of the last decade, sothe diHerent phases of the business cycle are well represented. Of course, we willbe mostly concerned with the performance at the end of the sample, but we arealso interested in assessing the sensitivity of the results to the state of the businesscycle.Hence, starting from January 1980, each of the models of the previous section is

estimated and 1–12 step-ahead forecasts are computed. Then, the forecast origin ismoved one step forward and the process is repeated until the end of sample is reached;note that we reestimate the model each time the forecast origin is updated, and soparameter estimation will contribute as an additional source of forecast variability.The experiment provides in total 252 one step ahead forecasts and 240 12-step-aheadforecasts.As hinted before, our benchmark model will be LLM; as the irregular variance was

always estimated equal to zero, yt is simply a random walk and thus its forecasts willbe used as a reference for comparison. Furthermore, we also included in our comparisonthe monthly ARIMA model considered by Montgomery et al. (1998); the latter is aseasonal model with orders (2; 0; 1)×(1; 0; 1)12 and its estimation with the full samplegave the following results:

(1− 1:88L + 0:88L2) (1− 0:53L12)yt = 0:001 + (1− 0:74L) (1− 0:80L12)�t ;

(0:05) (0:05) (0:07) (0:003) (0:07) (0:05);


Table 2Linear models: comparison of forecast performance in the test period 1980.1–2000.12

LLM LLTM TpCM CTM ARTM ARIMA

Lead time Mean error1 month −0:0091 −0:0014 −0:0117 −0:0054 −0:0046 −0:01441 quarter −0:0279 −0:0054 −0:0371 −0:0184 −0:0161 −0:04972 quarters −0:0684 −0:0229 −0:0904 −0:0565 −0:0500 −0:12393 quarters −0:1146 −0:0472 −0:1515 −0:1050 −0:0932 −0:21471 year −0:1579 −0:0698 −0:2101 −0:1518 −0:1351 −0:3090

Lead time Symmetric mean absolute percentage error1 month 1.87 1.84 1.96 1.83 1.79 1.761 quarter 3.44 3.26 3.89 3.20 3.09 3.142 quarters 5.83 5.91 7.33 5.56 5.09 5.573 quarters 8.00 8.95 10.95 7.83 6.93 8.131 year 10.24 12.57 14.50 10.29 9.09 11.02

Lead time Median relative absolute error1 month 1.00 1.00 1.11 1.00 0.97 0.931 quarter 1.00 0.86 1.30 0.98 0.90 0.942 quarters 1.00 0.90 1.52 0.94 0.85 0.953 quarters 1.00 1.02 1.69 0.97 0.85 1.101 year 1.00 1.07 1.76 0.99 0.87 1.21

Lead time Mean square forecast error1 month 0.0265 0.0247 0.0264 0.0237 0.0233 0.02211 quarter 0.1062 0.0928 0.1105 0.0811 0.0779 0.07472 quarters 0.3040 0.3276 0.3736 0.2416 0.2265 0.22473 quarters 0.5694 0.7868 0.8060 0.4887 0.4553 0.47891 year 0.8943 1.5627 1.3825 0.8266 0.7710 0.8393

The smallest values for each lead time are underlined.

where the sample variance of the innovation series �t is 0.0364. This is virtuallyidentical to that reported in the referenced paper, apart from the constant term, whichis not signi0cant; note that the nonseasonal AR polynomial contains a unit root. TheIRF is reported in Fig. 2; its pattern suggests that the model is capturing not solelythe residual seasonal eHect, but it is boosting the impact of innovations at short leadtimes; that the model is picking up something else is suggested by Montgomerty etal. in their comment to the quarterly version at the beginning of Section 3.1 of theirpaper. The implied persistence is 0.92, but the IRF makes a long and abrupt swingbefore converging to it.Table 2 reports a few basic statistics upon which forecasting accuracy will be as-

sessed. Denoting the l-step-ahead forecast for model (j) by y( j)t+l|t , we present the

average of the forecast errors (mean error), yt+l− y( j)t+l|t ; the symmetric mean absolute

percentage error (sMAPE), given by the average of 100|yt+l−y( j)t+l|t |=[0:5(yt+l+y( j)

t+l|t)],a measure which is featured in the M3-Competition (Makridakis and Hibon, 2000)and aims at providing a symmetric treatment of underforecasts and overforecasts; the


1 2 3 4 5 6 7 8 9 10 11 12

0.8

1.0

1.2

1.4

1.6

1.8 Relative MSFE

LLTM

TpCM

ARTMCTM

ARIMA

LLTMCTM ARIMA

TpCMARTM

1980 1985 1990 1995 2000

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Relative 1_step_ahead FMSE

Series ARTM

Fig. 3. Rolling forecast comparison of linear models: mean square forecast error relative to benchmark(LLM).

median relative absolute error (mRAE, see Armstrong and Collopy, 1992), a robustcomparative measure of performance computing the median of the distribution of the ra-tios |yt+l−y( j)

t+l|t |=|yt+l−y(1)t+l|t |, where (1) indexes the benchmark model; 0nally, we re-

port the mean square forecast error (MSFE). For simplicity these statistics are reportedonly for 1, 3 (1 quarter), 6 (2 quarters), 9 (3 quarters), and 12 (1 year) step ahead.In terms of MSFE the greatest accuracy is provided by the ARIMA model for

horizons up to two quarters; then its performance rapidly deteriorates and the bestforecasting model turns out to be ARTM. The latter is ranked best in terms of mRAEand sMAPE, immediately followed by CTM. Therefore, on aggregate, a simple struc-tural time series model, such as ARTM, is capable of outperforming the benchmarkARIMA model in Montgomery et al. (1998).The empirical evidence speaks strongly against TpCM, which is outperformed by

the naGSve model at all lead times. The table also tells that the forecasts are negativelybiased; the largest biases are found for the ARIMA model. LLTM presents a distinctivetrade-oH between bias and forecast error variance; as the forecast function is highlyadaptive, assigning a large weight to the most recent observations, the bias is verysmall, but on the other hand the forecasts’ variability is quite high.The left panel of Fig. 3 displays the MSFE of the linear models relative to that of

the benchmark model for every lead time. Among the structural models, the greatestaccuracy is provided by ARTM; yielding a proportional reduction in MSFE above 20%


for lead times greater than 2 months ahead. In conclusion, our preferred linear structuralmodel is ARTM; this will also represent the benchmark against which nonlinear models,dealt with in Section 5, are evaluated.In order to assess the cyclical sensitivity of its forecasting performance, we present in

the second panel the one-step-ahead relative MSFE for a rolling window of 60 consec-utive observations. The plot reveals that the forecasting accuracy deteriorates in periodsof slow decline in the unemployment rate (plotted in the background) and scores bet-ter in periods of rapidly rising and declining unemployment rates, during which thepersistence of the innovations should be greater. Similar behaviour characterises therelative MSFE of multistep forecasts. We mention in closing this section that the endof sample performance of the ARIMA model is worse than the naGSve model, with a10% increase in MSFE with respect to the benchmark.

4. Leave-k-out diagnostics and nonlinearity

Departure from normality was detected for the innovations of the ARTM; the break-down into the contribution of the two terms skewness and kurtosis reveal that the latteris 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). Theseare estimators of the disturbances associated with the components, conditional on theentire information set, e.g., E(�t |YT ), where �t is the cyclical disturbance in (4), andare computed from the output of the smoothing 0lter, reviewed in Appendix A.2, asindicated in Koopman (1993). Unlike the innovations, they are autocorrelated even ifthe model is correctly speci0ed, and Harvey and Koopman (1992) show how they canbe employed to form appropriate tests of normality, correcting for serial correlation.The skewness test conducted on E(�t |YT ) is highly signi0cant; signi0cant kurtosis wasalso detected.Additional evidence can be gathered using leave-k-out diagnostics, arising from the

deletion of groups of observations (see e.g., Bruce and Martin, 1989); these aim atspotting patches of observations that are not adequately 0tted by a linear model; forlinear state space models they are easily computed using the KF based algorithm pro-posed in Proietti (2000) and outlined in Appendix A.2. The case for computing them isquite strong in our application since dynamic asymmetries aHect clusters of consecutiveobservations.Fig. 4 is a plot of the indicator variable I(!(I) ¿c), !(I) being the statistic (A.5) in

Appendix A.2 and c the 5% critical value of the reference distribution F(k; T −k). Thedimension of the set of deleted observations is 1, 5, and 11; when k = 1 the statisticis a test for the presence of an additive outlier at time t; for k = 5; 11, the indicatorvalues refer to the midpoint of the deletion interval.As concerns leave-1-out diagnostics (upper panel), quite a few isolated observations

are Ragged in the 0rst year of the sample period up to 1961, and in the years 1974–1985; there is some clustering around turning points (especially noticeable aroundOctober 1949), in periods of fast rising unemployment (end of 1953, 1974). We suspect


1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000

0.5

1

Leave-1-out

1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000

0.5

1

Leave-5-out

1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000

0.5

1

Leave-11-out

Fig. 4. Leave-k-out diagnostics for ARTM.

that some masking has taken place, in that the eHect of adjacent observations could beclouded, and that joint deletion of consecutive observations can bring to the surfacethe masked outliers.As a matter of fact, the central and lower panels point out more clearly that viola-

tion of linearity and Gaussianity arises in economic downturns, that is in periods ofrapidly increasing unemployment, and around turning points. The other relevant pieceof evidence is that these departures are concentrated in the pre-forecast period; onlythe beginning of the test period, i.e. the slowdown at the beginning of the eightiesappears problematic. We anticipate that the gains in forecast accuracy from using anonlinear model will be small.


It must be acknowledged that in interpreting these plots a balance has to be madebetween unmasking (which is likely to have taken place during the downturns takingplace in the second-half of 1974 and in the 0rst-half of 1980) and smearing (probablyoccurring in 1949 and 1959): moreover, use of critical values from the F distributionwould be correct only if we knew the exact timing of the outlying eHect. Nevertheless,we 0nd these plots quite informative as a descriptive device for detecting groups ofsuspect observations.If estimation of the ARTM model is carried out considering as missing the observa-

tions producing signi0cant leave-5-out diagnostics, as suggested by one of the referees,only the irregular variance, 2

� , is aHected (the estimate value being 0.0085), whereasthe remaining parameter estimates are remarkably stable. The normality statistic isdrastically reduced, taking the value 34.45.

5. Nonlinear cyclical trend models

In this section, we consider four nonlinear trend models, derived from ARTM andCTM, that can account for dynamic asymmetries in unemployment rates, such as thosearising when unemployment rates are characterised by steep increases during recessionsand slower declines during expansions. The models belong to the class of smoothtransition structural time series models (see Proietti, 1999), according to which we letthe fundamental parameters vary according to the state of the system, as described byan appropriate transition variable.Hereby, we introduce the four alternative speci0cations and we report estimation re-

sults based on the full sample. The next section addresses the issues whether modellingasymmetries has some bearing on forecasting accuracy.

5.1. Speci5cation

In the ARTM case, for which the trend has the representation �t+1 = �t + t , with t+1 = � t + �t , it appears a sensible option to allow the autoregressive coeOcient �and the variance of the disturbances �t to evolve over time; thus, for a given measureof the state of the economy, St , bounded between 0 and 1, we set

�t = �0(1− St) + �1St ; 2�t = 2

�0(1− St) + 2�1St : (5)

The regime St is a function of a transition variable, zt , which is observable at time t; itschoice will be discussed in a moment. We will adopt a logistic transition mechanism(see e.g., Granger and TerGasvirta, 1993), de0ning St =[1+exp(−!(zt − c))]−1 where !is a smoothness parameter determining the speed of the transition, and c is a threshold.The resulting model is labelled ARTMSt.As far as CTM is concerned, it makes sense to allow variation also in the frequency

of the cycle, since it may be the case that expansions last longer than recessions; thiswould result in a lower cycle frequency in periods of declining unemployment. For


this purpose, we specify

�ct = �c0(1− St) + �c1St (6)

and denote the corresponding model CTMSt.Although including a drift term in the trend was not signi0cant globally (as hinted in

footnote 2), it might be the case that this plays a role locally. so we will also considerthe trend model:

�t+1 = �t + �t + t ; �t = �0(1− St) + �1St ;

where the AR and cycle parameters follow (5) and (6). This will give rise, respectively,to the ARTMStD and CTMStD models.The transition variable is usually based upon the diHerences %ryt = yt − yt−r , an

unweighted sum of current and past one-step changes in the unemployment rates; thishas two drawbacks: the transition is not necessarily smooth unless r is large; the dif-ferencing 0lter induces a phase shift, aHecting the timing of turning points. Anotherpossibility is to de0ne zt in terms of the underlying trend, e.g., zt=E[%r�t |Yt−1]=E[ t+· · ·+ t−r+1|Yt−1]; however, on the one hand, we do not expect great gains for a rela-tively smooth series such as the one considered here, and, on the other, inference wouldbe complicated computationally by the need to support the KF equations with a 0xedinterval smoother in order to construct the transition variable; if the analysis were con-ducted on seasonally unadjusted data, this would nevertheless be our preferred strategy.Our de0nition of zt builds upon the truncated version of the current-depth-of-recession

variable, proposed originally by Beaudry and Koop (1993) for US GNP, and laterused by Parker and Rothman (1997) to assess the role of business cycle asymme-tries in forecasting the US unemployment rate. The latter include (lagged values of)minj=0; :::; ryt−j − yt as an explanatory variable in an ARMA framework, measuringthe distance of the current rate from its historical local minimum; on the other handmaxj=0; :::; ryt−j − yt would measure the depth of the decrease in unemployment rates(an expansion in the economy).We propose to combine the two into the variable:

zt = 2yt − minj=0;:::;r

yt−j − maxj=0;:::;r

yt−j:

For monotonic patterns zt is coincident with %ryt , since either yt = minj=0; :::; ryt−j oryt =maxj=0; :::; ryt−j. When yt−1 is a turning point, zt = %ryt +Yyt = %(1 + Sr(L))yt ,so for instance after a peak has taken place, zt will be smaller than %ryt , as we addthe negative change from time t − 1 to time t; this results in an earlier recognitionof the change in regime and lessens the problem of phase shifts. The following tableillustrates the comparison for the case r=5 with reference to the pattern of observationsaround the turning point in December 1982 (yt = 10:8):

yt 9.6 9.8 9.8 10.1 10.4 10.7 10.8 10.4 10.4

%5yt 1.0 1.0 0.8 0.8 1.1 1.1 1.0 0.6 0.3zt 1.0 1.0 0.8 0.8 1.1 1.1 1.0 0.1 −0:1


Table 3Parameter estimates (Est.), standard errors (SE) and diagnostics for nonlinear structural models of US un-employment (“conc” denotes that the parameter is concentrated out of the likelihood function)

ARTMSt ARTMStD CTMSt CTMStD

Est. SE Est. SE Est. SE Est. SE

2�0 0.0029 0.10 0.0026 0.10 0.0009 0.04 0.0004 0.02

2�1 0.0643 2.24 0.0668 1.59 0.0550 1.91 0.0639 1.32

�0 0.8148 0.01 0.7867 0.01 0.8981 0.06 0.8957 0.03�1 0.4978 0.07 0.2289 0.03 0.5494 0.08 0.2095 0.05�c0 0.1419 0.02 0.1571 0.02�c1 0.0000 — 0.0000 —�0 −0:0160 0.03 −0:0193 0.01�1 0.1025 0.02 0.1172 0.02! 33.85 30.61 50.85 34.18 39.94 30.61 52.67 25.94c 0.3436 0.05 0.3204 0.02 0.3122 0.05 0.3127 0.022� 0.0097 conc 0.0099 conc 0.0109 conc 0.0116 conc

loglik 165.98 172.31 172.13 181.26Q(12) 25.51 23.59 19.70 21.36Q(24) 44.53 42.10 36.92 37.26Normality 125.78 86.17 122.02 62.48

Estimation period: 1948.1–2000.12.

When zt is coupled with the logistic transition mechanism it yields the regime vari-able St = [1 + exp(−!(zt − c))]−1; this framework is suitable for modelling businesscycle asymmetries such as those arising when unemployment rates are characterised bysteep increases during recessions (St close or equal to one) and slower declines duringexpansions (St close or equal to zero).

5.2. Estimation and testing

A score test of linearity against smooth transition alternatives can be carried out forunobserved components models following the strategy, proposed by Luukkonen et al.(1988), to circumvent the lack of identi0ability under the alternative; this amounts toreplacing St by a 0rst-order Taylor approximation around !=0 (see Proietti, 1999). Assuggested by TerGasvirta (1994), r is chosen by computing the score test for a set ofvalues and selecting the one yielding the smallest p-value. When applied to our datathe test is highly signi0cant and suggests r = 5; it must be stressed that, contrary toexpectations, the likelihood is also very informative on the choice of this parameter,displaying a distinctive peak at r = 5.As far as estimation is concerned, the resulting model is conditionally Gaussian

(see Harvey, 1989, Section 3.7.1., and Lipster and Shiryaev, 1978, Chapter 11), sincethe system matrices Tt; dt ; Ht , in Eq. (A.1), see Appendix A.1, depend solely on theinformation available at time t; likelihood inference is thus made via the predictionerror decomposition. Parameter estimates, along with standard errors and diagnosticsare reported in Table 3.


1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000

0.00

0.25

0.50

0.75

1.00

c

Series Transition variable (z_t)

S_t

Fig. 5. Transition variable, zt , and estimated transition function, St = [1+ exp(−!(zt − c))]−1, for the modelCTMStD, 1948.1–2000.12.

The variance of the AR and cycle disturbances are much higher in an economicdownturn; correspondingly, the damping parameter is signi0cantly lower for the modelsincluding a drift term (ARTMStD and CTMStD); for the latter the drift term is positiveand signi0cantly diHerent from zero. Moreover, the estimated frequency is zero for thecyclical models.Thus, the parameter estimates associated with the diHerent regimes imply that pe-

riods of increasing unemployment are characterised by greater uncertainty and lowerpersistence, although the trend disturbances are propagated with the addition of a pos-itive drift. This provides only a very preliminary description of the models’ dynamicproperties, hiding some of essential features, as will be argued later in Section 7, wherewe provide a more informative characterisation by means of the generalised impulseresponse function. Note that the in-sample goodness of 0t statistics are generally betterthan the linear case and that the normality statistic is drastically reduced. The modelsentertained are not suitable to deal with turning point asymmetries such as that locatedat 1949.10. If two interventions are added to the CTMStD model to account for twoadditive outliers occurring in October 1949 and February 1960 the normality statisticreduces to an insigni0cant 3.12.The estimated transition function, St , for CTMStD and the transition variable zt with

r = 5 are represented in Fig. 5.


6. Forecasting with nonlinear structural models

One-step-ahead forecasts are immediately available from the KF output at the endof the pre-forecast period; multistep forecasts, conditional on the estimated parameters,are generated by the Monte Carlo method (see Granger and TerGasvirta, 1993, Chapter8) as

y t+l|t =1M

M∑i=1

y(i)t+l|t ;

which, for i = 1; : : : ; M , requires the following steps:

1. draw y(i)t+1 ∼ f(yt+1|Yt)

2. draw y(i)t+2 ∼ f(yt+2|Yt; y

(i)t+1)

......

...l− 1. draw y(i)

t+l−1 ∼ f(yt+l−1|Yt; y(i)t+1; : : : ; y

(i)t+l−2)

l. evaluate y(i)t+l|t = E(yt+l|Yt; y

(i)t+1; : : : ; y

(i)t+l−1)

These steps are easily carried out with the support of the Kalman 0lter. The 0rst drawis made from the normal distribution N(y t+1|t ; FT+1), where the moments are obtainedfrom the last run of the KF within the sample (see Appendix A.1); then, conditional onz(i)t+1 and S(i)

t+1, one-step-ahead forecasts of the states and the observations are computedalong with their covariance matrices, which are in turn used to compute the next draw,y(i)t+2, and so forth.The predictive distribution, f(yt+l|Yt), can be estimated by the method of composi-

tion (see e.g., Tanner, 1996, Section 3.3), giving:

f(yt+l|Yt) =1M

M∑i=1

f(yt+l|Yt; y(i)t+1; : : : ; y

(i)t+l−1);

where the densities on the right-hand side are normal with mean Zt+la(i)t+l and variances

Ft+l delivered by the KF.The rolling forecast exercise is repeated for the four nonlinear models using M =

1000; the results are reported in Table 4, which for convenience reproduces thoseobtained for the linear benchmark (ARTM).The evidence is rather mixed; in terms of MSFE the linear benchmark is clearly

outperformed only by CTMStD, especially at short horizons; the best performancein terms of mRAE is provided by ARTMSt, instead. The latter and CTMSt tend toproduce positively biased forecasts and their overall performance is quite similar, asexpected. Somewhat unexpected is the poor performance of ARTMStD, compared toCTMStD; as a matter of fact, the latter proves rather unstable at the beginning ofthe pre-forecast period, yielding negative estimates of �1. This results in more volatileforecasts, especially in the beginning of 1980s, which are responsible for its poorcomparative performance in terms of MSFE.


Table 4Nonlinear models: comparison of forecast performance in the test period 1980.1–2000.12

ARTM ARTMSt ARTMStD CTMSt CTMStD

Lead time Mean error1 month −0:0046 0.0018 −0:0074 −0:0018 −0:01041 quarter −0:0161 0.0146 −0:0208 0.0082 −0:02542 quarters −0:0500 0.0346 −0:0565 0.0240 −0:06143 quarters −0:0932 0.0618 −0:0961 0.0469 −0:10271 year −0:1351 0.1028 −0:1270 0.0820 −0:1408

Lead time Symmetric mean absolute percentage error1 month 1.79 1.79 1.77 1.78 1.761 quarter 3.09 3.07 3.05 3.02 2.912 quarters 5.09 5.15 5.11 5.11 4.913 quarters 6.93 6.90 6.92 6.92 6.781 year 9.09 8.82 9.03 8.82 8.98

Lead time Median relative absolute error1 month 1.00 1.00 0.97 1.00 0.991 quarter 1.00 0.97 0.94 0.95 0.912 quarters 1.00 0.97 1.00 0.99 0.973 quarters 1.00 0.94 0.99 0.96 1.001 year 1.00 0.87 0.99 0.89 1.01

Lead time Mean square forecast error1 month 0.0233 0.0228 0.0226 0.0223 0.02171 quarter 0.0779 0.0821 0.0794 0.0793 0.07152 quarters 0.2265 0.2544 0.2385 0.2476 0.21343 quarters 0.4553 0.5045 0.4780 0.5005 0.44541 year 0.7710 0.8198 0.7931 0.8233 0.7675

Lead time Meese–RogoH test (p-values)1 month 0.24 0.15 0.15 0.051 quarter 0.82 0.70 0.62 0.032 quarters 0.94 0.95 0.86 0.083 quarters 0.91 0.94 0.85 0.291 year 0.82 0.85 0.78 0.44

“Best” values are underlined.

The last panel of the table reports the p-values of the Meese–RogoH test of com-parative forecast accuracy (Meese and RogoH (1988) see also Diebold and Mariano(1995), Clements and Hendry (1998, Chapter 13), for further discussion). In brief, itsrationale is the following: the null hypothesis of equal forecast accuracy (in the MSFEsense) implies a covariance of zero between the sum (st) and the diHerence (dt) of theforecast errors produced by the maintained and the benchmark models; under the nullthe estimated covariance is asymptotically normally distributed with zero mean andstandard error that depends on the auto and cross covariances of st and dt ; the lattercan be consistently estimated by e.g., a Bartlett window and used to build a statistic


0 5 10

0.925

0.950

0.975

1.000

1.025

1.050

1.075

1.100

1.125Relative MSFE

ARTM

ARTMSt

CTMSt

ARTMStD

CTMStD

1980 1985 1990 1995 2000

0.8

1.0

1.2

1.4

1.6

1.8

2.0

3_steps_ahead relative MSFE

CTM2_StM Series

Fig. 6. Rolling forecast comparison of nonlinear models: mean square forecast error relative to benchmark(ARTM).

for H0 : Cov(st ; dt)=0 with standard normal distribution. The results indicate that onlyCTMStD shows signi0cant improvements up until 6 step ahead.The left panel of Fig. 6 plots the MSFE of the four nonlinear forecasting models

relative to that of the benchmark against lead time; the percentage reduction achievedby CTMStD is around 8% for short leads and converges to zero as l increases. Thepanel on the left is a plot of the 3-step ahead relative MSFE for this model calculatedover spans of 5 years, with the original series reproduced in the background; the plot ismeant to provide some insight on how the comparative forecast accuracy of this modelvaries with time. It is noticeable that CTMStD yields the most signi0cant improvementstowards the end of the test period (20% MSFE reduction for the span including the last5 years in the sample), during a phase of slowly declining unemployment rates. It isalso remarkable that the performance is rather weak in the early 1990s, in connectionto the last upswing and successive decline, which corresponds to the period when thebenchmark yields the best results.To corroborate the 0nding that CTMStD provides better forecasts at the end of

the sample period, Fig. 7 presents the l-step-ahead (l = 1; : : : ; 12) forecasts from theorigin 1999.12 using M = 5000 replicates, along with the 95% highest density re-gion, HDR0:05 = {yt+l|t :f(yt+l|Yt)¿f0:05}, where f0:05 is chosen such that p(yt+l ∈HDR0:05|Yt)=0:95. The latter is the smallest of all possible 95% forecast regions and isparticularly helpful in revealing asymmetry and multimodality in the forecast densities;


0 5 10

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5Observed meanLinear LCL Linear UCL

Fig. 7. CTMStD model forecasts, 2000.1–2000.12, highest density regions, observed values and upper andlower 95% con0dence limits for the linear benchmark ARTM.

see Hyndman (1996) for more details. Even though at the forecast origin the systemis in an expansionary state, there is a possibility that it enters and gets stuck in arecessionary pattern; this shows up in the multistep forecast densities as a secondarymode. The 0gure also reports the upper and lower 95% con0dence bounds for theARTM forecasts, showing that the nonlinear model has reduced forecast uncertainty.

7. Generalised impulse response function

The notion of a generalised impulse response function (GIRF) provides further in-sight into the dynamic properties of CTMStD. The underlying idea is that in a nonlinearsystem the impact of an innovation occurring at time t on the future path of the processdepends crucially upon the history of the process and the size of the innovation.According to the de0nition by Koop et al. (1996),

GIRF(l; �t ; Yt−1) = E[yt+l|�t ; Yt−1]− E[yt+l|Yt−1];

where the notation stresses the dependence on the lead time l, the size of the innovationand the history of the process. The GIRF is evaluated by the same stochastic simulationtechniques illustrated in the previous section; basically, for a given lead time, innovation�t , and history, we evaluate the two components E[yt+l|Yt−1] and E[yt+l|�t ; Yt−1],averaging out intermediate innovations and subtracting out the results; it is crucial


-2 -1.5 -1 -0.5 0 1.5 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2Economic downturn

-2 -1.5 -1 0 1.5 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2Upturn

0.5 1 -0.5 0.5 1

Fig. 8. Generalised impulse response function. Plot of GIRF(60; �t ; Yt−1) (persistence) versus �t (innovation)for the model CTMStD during an economic downturn (increasing unemployment rates, left panel) and duringexpansions (decreasing unemployment rates, right panel).

that in the simulations we use common random numbers for drawing the intermediateshocks.The GIRF was computed for the empirical innovations yt − E(yt |Yt−1) resulting

from CTMStD, and using the observed history for l equal to a large number, l= 60;GIRF(60; �t ; Yt−1) can be taken as a measure of the long run impact, or persistence, ofthe observed shock. We note in passing that the cumulation of the persistence valuesis an approximation of the Beveridge and Nelson (1981) trend in the series.Fig. 8 is a plot of GIRF(60; �t ; Yt−1) versus the innovation �t conditional on the

state of the economy as described by the variable St : during a period of increasingunemployment (left panel) innovations tend to be positive and highly persistent; thereis also a small subset of very persistent large innovations. In periods of decreasingunemployment (right panel) we 0nd a cluster of highly persistent shocks, which occurafter a turning point, when unemployment rates are more rapidly declining. Anothergroup of small shocks displays low persistence and is gathered around the unit line;these occur in periods of low decreasing unemployment. This fundamental asymmetryin the dynamics of the system could not be anticipated by a simple characterisationbased on the parameter estimates in the two regimes, but is a quite natural consequenceof nonlinearity and of the basic fact that the response of the system depends on itsposition at the time of a shock and the size of the latter.


1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000

0.5

0.75

1

1.25

1.5

1.75

2

2.25

Positive inn.Negative inn.

Fig. 9. Normalised GIRF(3;±2SD�; Yt−1), for the nonlinear model CTMStD; the index plot in the back-ground displays the values of the transition function St .

These dynamic features also cast some light on the interpretation of the forecastingperformance of the CTMStD model. In this case, we would be interested in estimatingthe GIRF at lead times not greater than 12; for this purpose we set l=3, de0ne positiveand negative shock equal to twice the standard deviation of the innovations, denotedSD�, and compute GIRF(3;±2SD�; Yt−1) for all t starting from 1949; as before wecondition on the observed histories Yt−1. Fig. 9 plots the normalised GIRF for bothpositive and negative shocks, where the normalisation is made dividing for the initialshock. A noticeable feature is that positive shocks have a larger impact in a recession,a 0nding which is consistent with that of Koop and Potter (1999); both positive andnegative shocks have larger impact immediately after a peak, when the system movesout of a recession. Moreover, during periods of slowly declining unemployment ratesthe impact is damped out and it is less than the original size.In conclusion, the impact of shocks varies in expansions, being far greater after

a turning point has taken place and lately declining sharply when unemployment de-creases more slowly; then it jumps to a higher stable value as a downturn is entered. Fi-nally, the model tends to behave like a random walk with a positive drift in recessions.The fact that CTMStD produces better forecasts at the end of the sample period can

be ascribed to this feature of the model, which in periods of slowly declining rateshas a reduced persistence (recall that ARTM, implying large persistence, did not outperform better than the random walk model at the end of the sample, see the secondpanel of Fig. 3).


8. Concluding remarks

This paper has investigated the out-of-sample performance of linear and nonlinearstructural time series models of the US unemployment rate by means of an extensiverolling forecast experiment using a test period made up of the last two decades.The 0rst conclusion is that our study corroborates the hysteresis hypothesis, as we

found that linear models characterised by higher persistence perform signi0cantly better.The best performance was provided by a trend plus irregular model with the trend spec-i0ed as an ARIMA(1,1,0) process, with a positive and high autoregressive parameter.Persistence is not a stable property, however, and a nonlinear speci0cation, imply-

ing that this feature changes over the phases of the business cycle, outperforms theselected linear model at least at short lead times. In particular, it produces more accu-rate forecasts in periods of slowly decreasing unemployment rate, which is the regimeprevailing at the end of the sample period.The accuracy gains, though signi0cant, are not overwhelmingly great. This could be

anticipated by simple diagnostics immediately available from the linear 0t, revealingthat nonlinear and non Gaussian features are more prominent in the pre-forecast period(1948–1979).The general conclusion is that structural time series models prove a very useful

forecasting tool: they are parsimonious and are formulated in terms of core parametersand components with direct interpretation, which renders them easily extensible toaccount for nonlinearity.

Acknowledgements

This research was supported by Co0n. MURST 2000—prot. MM13035581 003 aspart of the project Linearity and Nonlinearity in Time Series Dynamics. The paper waspresented at the Econometrics Seminars, Tinbergen Institute, Amsterdam, the Schoolof Finance and Economics, University of Technology Sydney, the 2001 EconometricSociety Australasian Meeting, University of Auckland, New Zealand, the EuropeanCentral Bank, and the SCO 2001 Conference in Brixen, Italy. I would like to thankthe participants, in particular Estela Bee Dagum, and two anonymous referees for theircomments.

Appendix A.

A.1. The Kalman 5lter

The models considered in the paper admit the state space representation:

yt = Zt1t + Gt�t ; t = 1; 2; : : : ; T;

1t+1 = Tt1t + dt + Ht�t (A.1)


with �t ∼ NID(0; I) and 11 ∼ N(a1; P1) independent of �t ;∀t. The model is timeinvariant if the system matrices Zt; Gt; Tt ; dt ; Ht , do not depend on t.The Kalman 0lter (KF) (Anderson and Moore, 1979), is a well-known recursive

algorithm for computing the minimum mean square estimator of 1t and its mean squareerror (MSE) matrix conditional on Yt−1 = {y1; y2; : : : ; yt−1}. De0ning at = E(1t |Yt−1),MSE(at) = Pt = E[(1t − at)(1t − at)′|Yt−1]; it is given by the set of recursions:

�t = yt − Ztat ; Ft = ZtPtZ ′t + GtG′

t ;

qt = qt−1 + �′tF−1t �t ; Kt = (TtPtZ ′

t + HtG′t)F

−1t ;

at+1 = Ttat + dt + Kt�t ; Pt+1 = TtPtT ′t + HtH ′

t − KtFtK ′t (A.2)

with q0 = 0, �t =yt −E(yt |Yt−1) are the 0lter innovations or one-step-ahead predictionerrors, with MSE matrix Ft . Initialisation of the state vector when nonstationary statecomponents are present is discussed in Koopman (1997).The Kalman 0lter is said to have reached a steady state if, for some t, Pt = P. The

conditions under which limt→∞Pt = P are given in Harvey (1989, Sections 3.3.3 and3.3.4), and, when they are met, the matrix P is the solution of the Riccati equationP = TPT ′ + HH ′ − KFK ′; with K = (TPZ ′ + HG′)F−1 and F = ZPZ ′ + GG′.

A.2. Leave-k-out diagnostics

Leave-k-out diagnostics are based upon the output of the smoothing 0lter (De Jong,1988, 1989; Kohn and Ansley, 1989):

ut = F−1t �t − K ′

t rt ; Mt = F−1t + K ′

t NtKt ;

rt−1 = Z ′t F

−1t �t + L′

trt ; Nt−1 = Z ′t F

−1t Zt + L′

tNtLt ; (A.3)

Lt = Tt − KtZt , started with rT = 0 and NT = 0; ut is termed a smoothing error.Assuming that observations yi−k+1; : : : ; yi are deleted, let us denote the stack of

the deleted observations by y(I) and the vector of deletion residuals by d(I) = y(I) −E[y(I) |y1; : : : ; yi−k ; yi+1; : : : ; yT ]; let also y = (y1; : : : ; yT )′ ∼ N(0; V ), so that qT =y′V−1y.The statistic

!(I) =d′

(I)[Cov(d(I) )]

−1d(I)

qT − d′(I)[Cov(d(I) )]

−1d(I)

· T − kk

:

provides a test that the observations are jointly outlying. Under normality the exactdistribution of !(I) is F(k; T −k). It can be computed by running backwards the Kalman0lter on the pseudo-model made up of the measurement equation ut = F−1

t �t − K ′t rt

and transition equation rt−1 =Z ′t F

−1t �t +L′

trt , where F−1=2t �t act as disturbances, ut are

the observations and rt the states. This produces:

u∗t = ut + K ′t r

∗t ; M∗

t = F−1t + K ′

t N∗t Kt ;


q∗t−1 = q∗t + u∗′

t M∗−1t u∗t ; K∗

t = (Z ′t F

−1t − L′

tN∗t Kt)M∗−1

t ;

r∗t−1 = L′tr

∗t + K∗

t u∗t ; N ∗

t−1 = Z ′t F

−1t Zt + L′

tN∗t Lt − K∗

t M∗t K

∗′t (A.4)

for t = i; i − 1; : : : ; i − k + 1. The 0lter is initialised by the unconditional mean andcovariance matrix of ri, that is r∗i = 0 and N ∗

i = Ni.In Proietti (2000) it is shown that q∗i−k+1 = d′

(I)[Cov(d(I) )]

−1d(I) so the statistic canbe computed as

!(I) =q∗i−k+1

qT − q∗i−k+1

T − kk

: (A.5)

Note that, once the 0lter (A.4) is run for the maximum k desired, leave-j-out diagnos-tics for j¡k are immediately available.

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Forecasting the US unemployment rate