Estimating persistence in Canadian unemployment: evidence from a Bayesian ARFIMA

Abstract

The degree of persistence in aggregate Canadian unemployment is estimated within a Bayesian ARFIMA class of models. The results conclude that unemployment exhibits persistence in the short and intermediate run. The evidence of persistence is stronger than previously reported by Koustas and Veloce (1996). This persistence cast a vital implication regarding disinflation policies, Based on the unemployment rate, these policies will prove very costly in terms of lost output and – if implemented – they considerably lengthen recessions.

Acknowledgements

The first author is grateful to the McGill Major Fellowship for financial support, to John W. Galbraith and to the seminar participants at the University of Central Florida for helpful comments.

Notes

¹ See Wu (2003) for a panel data estimation of unemployment persistence in China.

² In fairness to the literature, the estimation methods can be classified as follows, (1) Maximum Likelihood method (Sowell, 1992); (2) Approximate Maximum Likelihood method (Li and Mcleod, 1986, Baillie and Chung, 1993, or using the Whittle approximation as outlined by Fox and Taqqu, 1986), where the estimation of the fractional integration parameter ‘d’ is done at the same time as the estimation of the other parameters (coefficients of the AR and the MA parts); (3) Two-step procedures (Janacek, 1982 and Geweke and Porter-Hudak, 1983) and finally (4) The non-iterative approximation based estimators as in Durbin (1959) and Galbraith and Zinde-Walsh (1994). This method relies on approximating the moving average process by an autoregressive model and uses the pattern of autoregressive coefficients to deduce estimates of the parameters of the underlying process. The Galbraith and Zinde-Walsh estimator have a lower bias than Durbin's for a given approximation order. This class of estimators is asymptotically efficient and more robust–regarding misspecification – to maximum likelihood based methods.

³ The variable of interest is the log of total unemployment for Canada. The CANSIM label is D980739. The quarterly data covers the period from 1976 to 1999. This sample period is focused on for the following reasons. Prior to 1976, the Canadian government undertook an overhaul to the unemployment insurance benefits. Post 1999, CANSIM changed the labeling of the series.

⁴ Mikhail et al . (2005) tested for persistence in aggregate and sectoral Canadian unemployment and reported strong evidence of long-range dependence.

⁵Granger (1980) showed that the contemporaneous aggregation of panel data resulted in fractionally integrated processes. Therefore, given that total unemployment is a contemporaneous sum of N sectors unemployment, it is appropriate to treat the Canadian total unemployment – using Statistics Canada Input–Output Tables, the L-level of Canadian unemployment includes 112 industries – level data as following a long-memory process.

⁶ The HP filter ‘removes important time series components that have traditionally been regarded as representing business cycle phenomena’ King and Rebelo (1993). For a complete discussion of the negative effects of the Hodrick–Prescott, see Stadler (1994) and Cogley and Nason (1995). For spurious cyclical behaviour induced by the filter, see Harvey and Jaeger (1993). Using the HP filter, the results were computed and not reported for space.

⁷ In other words, the space of the fractional differencing parameter is restricted to d ∈ (0.0, 1.5). In the case where δ = 0, d equals 1 and the process y_t is modelled as an ARIMA (p, 1, q), i.e., z_t is an ARMA(p, q). The restriction on the space of δ merits some explanation. The lower bound of δ (−1) ensures that Δy_t is invertible (see Odaki, 1993). Also, from Fig. 2, the autocorrelations are positive and decay hyperbolically. Therefore, restricting the lower bound of d to zero is coherent. A reasonable implication of the unit root test is to restrict the upper bound to 0.5 which ensures that Δy_t is stationary. Whenever d ∈ (0.0, 0.5), y_t is said to be trend-stationary with long memory. Whenever d ∈ (0.5, 1.5), Δy_t is stationary with intermediate memory for d < 1 and with long memory for d > 1.

⁸ Since IMP(∞) is of little relevance to the economic forecaster, they defined persistence as follows. If the frequency of the data at hand is quarterly, and IMP(4), IMP(12) and IMP(40) are referred to as the short-run, medium-run and long-run impact of a shock respectively, then an economist is only interested in these quantities: IMP(4), IMP(12) and IMP(40). If ‘unemployment persistence’ is defined as ‘the effects lasting for at least two years’. Therefore, computing the suggested impulse responses will quantify persistence.

⁹ The antithetic replication is computed by projecting the draw through the mean of the uniform distribution [−1.0, 0.5], i.e., −0.25. Therefore, the antithetic value equals −0.25 − [draw − (−0.25)]. Formally, the antithetic value . See Dorfman (1997) for more details.

¹⁰ This exercise was repeated 25 000 times. The simulation was carried out on i686 machine running LINUX 2.2.14-5.0. The FORTRAN 77 code from Koop et al. (1997) is gratefully acknowledged. The code was modified to fit the problem. Conditional on the number of parameters in each model, the average time for simulating one model was 23 minutes.

¹¹ For the regularity conditions to insure the convergence of g(ω), see Dorfman (1997). The general criteria for choosing an importance function are discussed in Bauwens et al. (1999).

¹² Other methods of simulation such as the Gibbs sampling algorithm were not focused on. The Gibbs sampling algorithm has been used for the analysis of univariate ARFIMA processes by Pai and Ravishanker (1996, 1998).

¹³ The scope of Bayesian model comparison and model assessment is quite broad. The literature on Bayesian model comparison can be classified as (1) the marginal likelihood approach, (2) the ‘super-model’ or ‘sub-model’ approach and (3) the criterion-based methods such as the L measure and the calibration distribution (Chen et al. 2000). The second approach is efficient whenever the posterior means or modes are not far from zero. The last approach does not require proper prior distributions over the models. Here, the marginal likelihood approach is adopted. This approach is essentially the same as the Bayes factor approach.

¹⁴ The success of the Bayesian predictive distribution as a model checking device is discussed at length by Geisser (1993). This paper does not address the prediction aspect of the proposed model but focuses only on the persistence issue.

¹⁵ Using a squared error loss, mixing over the models is optimal for forecasting (see Min and Zellner (1993) for the proof).

¹⁶ Since all conceivable models for the problem at hand are not considered, model comparison based on the posterior odds does not change if a new unspecified third model is introduced. Given the intuitive economic argument, it is suggested that the Independence of Irrelevant Alternatives (IIA) property holds (see Poirier (1997) for the definition and for an a comprehensive exposition).

¹⁷ See Box (1980, p. 408), Carlin and Louis (1996, p. 47, Equation 2.17) and Chen et al. (2000, p. 237, Equation 8.1.3).

¹⁸ On the quantification of ignorance, see Bauwens et al. (1999). Briefly, the approach adopted here maximizes the entropy of the model density over the parameter space.

¹⁹ Common choices of loss function are: (1) the quadratic loss , (2) the absolute loss and (3) the zero-one loss L(δ, δ) = c if  ≠ δ and L(δ, δ) = 0 if  = δ. See Dorfman (1997) for the derivations. Choosing the quadratic (absolute, zero-one) loss results in the mean (median, mode) as the Bayesian optimal point estimate.

²⁰ Here, the symmetric ‘0 − K_i ’ loss function is adopted, as defined in Bauwens et al. (1999). The probability of errors of type I and II are equal. See also Zellner (1987) where ‘under a symmetric loss structure, a comparison of the posterior probabilities will provide a basis for choosing between H ₀ and H ₁.’