Testing an Autoregressive Structure in Binary Time Series Models

ömmföäflsäafaäsflassflassflas ffffffffffffffffffffffffffffffffffff Discussion Papers Testing an Autoregressive Structure in Binary Time Series Models Henri Nyberg University of Helsinki and HECER Discussion Paper No. 243 November 28 ISSN 795-562 HECER Helsinki Center of Economic Research, P.O. Box 7 (Arkadiankatu 7), FI-4 University of Helsinki, FINLAND, Tel +358-9-9-2878, Fax +358-9-9-2878, E-mail info-hecer@helsinki.fi, Internet www.hecer.fi

HECER Discussion Paper No. 243 Testing an Autoregressive Structure in Binary Time Series Models* Abstract This paper introduces a Lagrange Multiplier (LM) test for testing an autoregressive structure in a binary time series model. Simulation results indicate that the two versions of the proposed LM test have reasonable size and power properties when the sample size is large. A parametric bootstrap method is suggested to obtain approximately correct sizes also in small samples. The use of the test is illustrated by an application to recession forecasting models using monthly U.S. data. JEL Classification: C2, C25 Keywords: LM test, Binary response, Dynamic probit model, Parametric bootstrap, Recession forecasting Henri Nyberg Department of Economics University of Helsinki P.O. Box 7 (Arkadiankatu 7) FI-4 University of Helsinki FINLAND e-mail: henri.nyberg@helsinki.fi *The author would like to thank Markku Lanne and Pentti Saikkonen for constructive comments. The author is responsible for remaining errors. Financial support from the Academy of Finland, the Research Foundation of the OP Group and the Finnish Foundation for Advancement of Securities Markets is gratefully acknowledged.

Introduction Recently, Rydberg and Shephard (23), Chauvet and Potter (25) and Startz (28), among others, have introduced new time series models for binary dependent variables. In this paper, the dynamic autoregressive probit model suggested by Kauppi and Saikkonen (28) is considered. We develop Lagrange Multiplier (LM) tests which can be used to test the adequacy of a restricted model in which the autoregressive structure is excluded. The proposed LM tests are attractive because they only require estimates from the restricted models, which can be obtained by using standard econometric software packages. According to our simulations, the two versions of the LM test considered have reasonable size and high power, especially in large samples. In small samples, a parametric bootstrap method is proposed to obtain critical values which are more reliable than the asymptotic ones. In an empirical application, the LM tests are used to assess recession forecasting models for the U.S. The paper is organized as follows. The probit model is introduced in Section 2 and the LM tests are developed in Section 3. Results of the simulation and bootstrap experiments are provided in Section 4 and the empirical example is presented in Section 5. Finally, Section 6 concludes. 2 Model Consider the binary valued stochastic process y t, t =, 2,..., T, and let E t ( ) and P t ( ), respectively, signify the conditional expectation and conditional probability given the information set Ω t. Conditional on Ω t, y t has a Bernoulli distribution, that is, y t Ω t B(p t ). () In the probit model p t = E t (y t ) = P t (y t = ) = Φ(π t (θ)), (2) where Φ( ) is a standard normal cumulative distribution function. The model π t (θ) is a linear function of variables in the information set Ω t and the parameter vector θ.

In previous literature, the static model π t (θ) = ω + x t β, (3) has been the most commonly used specification. It has been employed in various applications, such as, forecasting the recession periods of an economy (see e.g. Estrella and Mishkin, 998). A natural extension of the static model (3) is the dynamic specification π t (θ) = ω + δ y t + x t β, (4) where the lagged value of the dependent variable is also assumed to belong to the information set. Kauppi and Saikkonen (28) generalize the dynamic model (4) by adding a lagged value of π t (θ), giving π t (θ) = ω + α π t (θ) + δ y t + x t β, (5) where α <. This induces a first-order autoregressive structure to the model equation. It is worth noting that alternative, but very similar, models have been proposed by Rydberg and Shephard (23) and Kauppi (28). The LM tests developed in the next section can straightforwardly be extended to these models as well. The parameters of the models (3) - (5) can conveniently be estimated by the method of maximum likelihood (ML). Conditional on initial values, the log-likelihood function is l(θ) = T l t (θ) = t= T t= ( ) y t log(φ(π t (θ))) + ( y t ) log( Φ(π t (θ))), (6) where l t (θ) is the log-likelihood for t:th observation. The score function is s(θ) = l(θ) T = s t (θ) = t= T ( t= y t Φ(π t (θ)) Φ(π t )( Φ(π t (θ))) φ(π t(θ)) π t(θ) ), (7) where φ( ) signifies the probability density function of the standard normal distribution and an explicit expression of the derivative term π t (θ)/ will be given in the next section. The ML estimator ˆθ, which solves the first order condition s(ˆθ) =, is found by maximizing the log-likelihood function (6) with numerical methods. For simplicity, only the first lags of the dependent variable y t and π t (θ) are employed in this paper. 2

3 LM Tests In applications, model (5) may be a superior to its restricted version (4) but, on the other hand, its ML estimation is more complicated and no estimation procedures are readily available in standard econometric software packages. Thus, it is of interest to start with the simpler model (4) and check for its adequacy by testing whether the autoregressive coefficient α in (5) is zero. The null hypothesis of interest is therefore H : α =. (8) In this context, the LM test is attractive because it only requires the estimation of the parameters of model (4). The general LM test statistic (see e.g. Engle, 984) for the null hypothesis (8) can be written as LM = s( θ) I( θ) s( θ), (9) where θ is the ML estimate of θ restricted by (8) and I( θ) is a consistent estimate of the information matrix I(θ). Under H, the test statistic (9) has an asymptotic χ 2 - distribution. Following Davidson and MacKinnon (984) we can construct two LM test statistics for the null hypothesis (8). The first one is ( ) LM = ι S( θ) S( θ) S( θ) S( θ) ι, () where ι is a vector of ones and the matrix S( θ) is given by S( θ) = ( ) s ( θ) s 2 ( θ)... s T ( θ). Expression () can also be seen as the regression sum of squares from the artificial linear regression ι = S( θ)a + error. Using the symbols Φ t = Φ(π t ( θ)) and φ t = φ(π t ( θ)), a second LM test statistic can be based on the artificial regression r( θ) = R( θ)b + error, () 3

where with and R( θ) = (R ( θ) R 2 ( θ)... R T ( θ) ) R t ( θ) = ( Φt ( Φ ) /2 π t ( θ) t ) φt r( θ) = ( ) r ( θ) r 2 ( θ)... r T ( θ) with ( ) Φt /2 ( ) Φt /2 r t ( θ) = y t + (yt ) Φ t Φ t = (( Φ ) /2 t ) Φ t (y t Φ ) t. Running the artificial regression () and computing the regression sum of squares yields the test statistic ( ) = r( θ) R( θ) R( θ) R( θ) R( θ) r( θ). (2) Because R( θ) r( θ) = s( θ) = S( θ) ι, it can be seen that the test statistics LM and only differ in the way the information matrix estimate I( θ) is constructed. and Note that the test statistics LM and can also be expressed as = where LM = T t= T t= ( πt ( θ) ) ( T d t t= ( πt ( θ) ) ( T d t t= d 2 t( πt ( θ) φ 2 t Φ t ( Φ t ) d t = )( πt ( θ) ) ) T t= ( πt ( θ) y t Φ t Φ t ( Φ φ t. t ) )( πt ( θ) ) ) T t= ( πt ( θ) ) d t, ( πt ( θ) ) d t, This shows that the derivative term π t (θ)/ evaluated at θ is central for the test statistics. From (5), the derivative is defined as π t (θ) = π t(θ) ω π t(θ) α π t(θ) δ π t(θ) β = 4 + α π t (θ) ω π t (θ) + α π t (θ) α y t + α π t (θ) δ x t + α π t (θ) β

and π t ( θ) = π t( θ) ω π t( θ) α π t( θ) δ π t( θ) β = π t ( θ) y t x t. 4 Simulation Results The two LM tests described in the previous section are asymptotically equivalent. In this section their finite small-sample properties are studied by simulation. 2 We simulated realizations from the Bernoulli distribution () using two different models π t (θ) =.3 + α π t (θ) +.5 y t (3) and π t (θ) =.3 + α π t (θ) +. y t x t. (4) Since many macroeconomic and financial time series exhibit rather strong persistence we assume the following AR() process for the explanatory variable x t, x t =. +.9x t + ε t, ε t NID(, ). Positive coefficients for the lagged y t in (3) and (4) indicate that the realized values of y t, i.e. zeros and ones, tend to cluster in the same way as, for example, recession periods of the economy (see Section 5). We provide simulation evidence for sample sizes 5, 3, 5, and 2. For all generated series, 2 extra observations were simulated and discarded from the beginning of every sample to avoid initialization effects. We report empirical sizes of the models at %, 5% and % significance levels. All results are based on 2 replications. However, in some cases a little more than 2 replications are needed because of numerical difficulties in the optimization of the log-likelihood function (6). Empirical sizes of the LM tests for selected parameter values in (3) and (4) are presented in Tables and 2. The empirical sizes range between sample sizes. Both 2 Matlab version 7.5. is used in simulation and estimation. Eviews code for computing LM tests () and (2) is also available upon request. 5

tests seem to be rather severely oversized in small samples (T = 5, T = 3 and T = 5), but for larger samples, the empirical sizes are rather close to the nominal levels, especially at the 5 % level. Table : Empirical size of the LM and tests in the model (3). T LM % 5% % % 5% % 5 2.7 4.6 7.2 2.7 4. 6.6 3 2.9 8.4 3.7 3. 8.3 3.6 5 2. 8. 3.7 2. 8. 3.8. 6.2 3. 6.3 3. 2 7. 5. 2.6 7. 5. 2.6 Notes: In size simulations, α =. The results are based on the 2 replications. Table 2: Empirical size of the LM and tests in the model (4). T LM % 5% % % 5% % 5 33.9 25.6 4.3 3.5 2.9.3 3 2.8 4.7 7.5 9.7 3.9 6. 5 5.3.4 4.8 4.4. 4..4 7. 3.3 7. 3.2 2 7.7 5.6 2.4 7.5 5.2 2.4 Notes: See notes to Table. Rejection rates presented in Tables and 2 are based on the critical values from the asymptotic χ 2 - distribution. However, one can use a parametric bootstrap method to obtain alternative, potentially more accurate, critical values than the asymptotic ones. The employed procedure is the following. ML estimates θ = (ˆω ˆδ ˆβ) and LM test statistics are computed under the null hypothesis α =. Bootstrap samples y b τ and the values of test statistics LM b and LMb 2, b =, 2,..., B, are then generated 6

from the data-generating process where τ =, 2,..., T, and y b τ B(Φ(π b τ( θ))), (5) π b τ( θ) = ˆω + y b τ ˆδ + x t ˆβ, Finally, bootstrap critical values at different significance levels are obtained from the empirical distribution of the test statistics LM b and LMb 2. The number of bootstrap replications B is set to 5 and the simulation is carried out for 5 replications. As an illustration for the usefulness of the proposed bootstrap method, Table 3 presents the rejection rates based on the bootstrap critical values instead of the asymptotic ones. Compared with the results shown in Table 2, the empirical sizes of the LM tests are now much closer to the nominal values. Table 3: Empirical size of the LM and tests using the model (4) and bootstrap critical values. LM T % 5% % % 5% % 5 9. 4.2. 9.6 6.2. 3 9.6 5.4.6 9. 6.2. 5 2.2 5..2.2 6.4.4 Size-adjusted empirical power functions at the 5 % level and all examined sample sizes T are depicted in Figures and 2 using (3) and (4) with different values of α. For simplicity, we expect that the sign of the parameter α is non-negative and therefore concentrate on values from α =. up to α =.8. The power seems to increase rather quickly when the value of α increases, in particular when the explanatory variable x t is employed in the model. It appears that the power of is typically slightly higher that of LM in both cases. However, the differences are not very large. Even for smaller sample sizes 5, 3 and 5, reasonable power is obtained although the power of tests is slightly decreasing at very high values of α. As Kauppi 7

and Saikkonen (28) note, a potential reason for this finding is that π t (θ) and y t may interact in a complicated way which could affect the statistical significance of the π t (θ), especially in small samples. T=5 T=3.9.9.8.8.7.7.6.6.5.5.4.4.3.3. LM. LM.4.6.8.4.6.8 α α T=5 T=.9.9.8.8.7.7.6.6.5.5.4.4.3.3. LM. LM.4.6.8.4.6.8 α α T=2.9.8.7.6.5.4.3 LM..4.6.8 α Figure : Empirical power in the case of model (3). 8

T=5 T=3.9.9.8.8.7.7.6.6.5.5.4.4.3.3. LM. LM.4.6.8.4.6.8 α α T=5 T=.9.9.8.8.7.7.6.6.5.5.4.4.3.3. LM. LM.4.6.8.4.6.8 α α T=2.9.8.7.6.5.4.3. LM.4.6.8 α Figure 2: Empirical power in the case of model (4). 9

5 Application: U.S. Recession Forecasting Models Forecasting recession periods has been one of the most common empirical applications of binary time series models. Predicting the direction-of-chance in stock market returns is an example of another potential application (see e.g. Rydberg and Shephard, 23). In recession forecasting the dependent variable is a recession indicator, if the economy is in a recessionary state at time t, y t =, otherwise. Although in this study we are not interested in out-of-sample forecasting, we consider forecasting models behind the direct (using horizon-specific y t 5 ) and iterative (y t ) multi-step forecasts for the binary response (6) (for details, see Kauppi and Saikkonen, 28). The difference between direct and iterative forecasts is similar to that in time series models for traditional continuous variables (see e.g. Marcellino, Stock, and Watson, 26). (6) In this study the recession periods identified by the National Bureau of Economic Research (NBER) for the United States between April 953 and December 26 are employed. Several studies have shown that the term spread (SP t ) between the longterm and short-term interest rate is a useful predictive variable. In addition, the stock market return (r t ) has also been found to have predictive content (see e.g. Estrella and Mishkin, 998 and Nyberg, 28). Table 4 presents the estimated predictive models when the forecast horizon h is assumed to be six months (h = 6). The fact the NBER business cycle turning points are announced with a delay is taken into account in direct forecasting models. 3 In a direct forecasting model shown in the first column of Table 4, the p-values of the two LM tests based on asymptotic critical values are zero indicating that the inclusion of an autoregressive structure gives a better model. The same conclusion is drawn by using bootstrap critical values. Further, when π t (θ) is included in the model, it is a statistically significant predictor (model 2) according to the Wald-type 3 We assume that this publication lag is nine months. For further details see e.g. Kauppi and Saikkonen (28), Kauppi (28), and Nyberg (28).

Table 4: Estimation results for the recession prediction models. model 2 3 4 constant -.5 -.2 -.7-2. (.6) (.4) (.7) (4) SP t 6 -.6 - -.58 -.67 (.3) (.5) (.4) (.6) r t 6 -.5 -.8 -. -. (.2) (.2) (.3) (.3) π t (θ).8 -.7 (.3) (.9) y t 3.38 3.95 (5) (.35) y t 5 -.4 -.7 (.36) (.6) log-l -85.94-36.6-55.63-55.29 pseudo R 2.92.367.689.69 LM 25.25 2.75 p-value..69 36.8.65 p-value..746 Bootstrap critical values LM % 3.6 5. 5 % 3.93 6.88 % 5.9 9.6 % 2.7 2.3 5 % 3.75 3.7 % 5.37 6.45 Notes: Models are estimated using U.S. data from 953 M4 to 26 M2 (T = 627). First 8 months are used as initial values. Robust standard errors (see Kauppi and Saikkonen, 28) are reported in parentheses. The estimated value of the log-likelihood function (6) and pseudo-r 2 measure (Estrella, 998) are also provided as well as the values of the LM and test statistics, their p-values based on the asymptotic χ 2 -distribution, and critical values obtained by bootstrap.

of test comparing the estimated coefficient and its standard error or using likelihood ratio test between the competitive models. The values of estimated log-likelihood function and the pseudo-r 2 measure (Estrella, 998) are also substantially higher in the latter model. When the iterative predictive model 3 is considered, the values of the test statistics LM and are statistically insignificant at traditional significance levels compared with both asymptotic and bootstrap critical values. When the autoregressive structure is imposed on the model the coefficient for π t (θ) is indeed statistically insignificant in the extended model 4. Interestingly, the estimated value of the parameter α is negative. In conclusion, in these two examples, outcomes of the two LM tests are in accordance with the Wald test when testing the statistical significance of the autoregressive structure. The recommendation is that an autoregressive model structure is worth considering as an alternative to a static recession prediction model (see e.g. Estrella and Mishkin, 998), possibly augmented by the forecast horizon-specific lagged value of y t, as presented in the first model of Table 4. However, in iterative forecasting model the lagged state of the economy, y t, seems to be the main dynamic part of the model. 6 Conclusions We have proposed LM tests for testing an autoregressive model structure in binary time series models. Based on a limited simulation study, the tests appear to have reasonable empirical size, especially in large samples, and high power. For small samples, the proposed bootstrap simulation method provides improved empirical sizes. An empirical example of recession forecasting models in the United States illustrates that the inclusion of an autoregressive model structure may be a useful addition to the recession prediction model. 2

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