On Joint Determination of the Number of States and the Number of Variables in Markov-Switching Models: A Monte Carlo Study

Abstract

In this article, we examine the performance of two newly developed procedures that jointly select the number of states and variables in Markov-switching models by means of Monte Carlo simulations. They are Smith et al. (2006) and Psaradakis and Spagnolo (2006), respectively. The former develops Markov switching criterion (MSC) designed specifically for Markov-switching models, while the latter recommends the use of standard complexity-penalised information criteria (BIC, HQC, and AIC) in joint determination of the state dimension and the autoregressive order of Markov-switching models. The Monte Carlo evidence shows that BIC outperforms MSC while MSC and HQC are preferable over AIC.

Keywords:

• Information criteria
• Markov-switching model
• Monte Carlo

Mathematics Subject Classification:

• 62J02
• 62M05
• 65C05

Notes

¹In fact, Smith et al. (2006) made the following remark on p. 560: “We encourage further research to investigate such comparisons using the proposed MSC.”

²Autoregressive model is used here for illustration purpose only. To get the equation for static model, just replace the set of lagged y _t variables by the same number of exogenous variables, x _t .

³We would like to thank an anonymous referee for suggesting this persistence scenario for us to examine.

The data generating process is the MS static model with two regimes and one exogenous variable shown as follows:

The initial probabilities are set to 0.4 and 0.6 with the transition probabilities as p ₁₁ = 0.6; p ₂₂ = 0.4. The true regression coefficients are v ^{(s _t )} = (v ⁽¹⁾, v ⁽²⁾) and A ^{(s _t )} = (A ⁽¹⁾, A ⁽²⁾), where v ^{(s _t )} = (0, 0) and A ^{(s _t )} = (0.3, 0.9). Notice that the total replication is 1,000.

The first column shows the possible values that σ^{(s _t )} can take. Notice that in the regime-specific case, σ^{(s _t )} is set to 0.5 if s _t  = regime 1 and 1 if s _t  = regime 2. The second column shows the relevant information criterion being considered, while the third column indicates the number of regimes imposed on estimation. With sample size of 100, the fourth column depicts the frequency of selecting the correct number of variable (in this case, 1 variable) while the fifth column indicates the frequency of selecting the incorrect number of variables (in this case, 2 variables). Other columns show similar frequencies for different sample sizes.

The true data generating process is the MS static model with two regimes and one exogenous variable, as follows:

The initial probabilities are set to 0.4 and 0.6 with the transition probabilities as p ₁₁ = 0.6; p ₂₂ = 0.4. The true regression coefficient coefficients are v ^{(s _t )} = (v ⁽¹⁾, v ⁽²⁾) and A ^{(s _t )} = (A ⁽¹⁾, A ⁽²⁾), where v ^{(s _t )} = (0, 1) and A ^{(s _t )} = (0.3, 0.3). Notice that the total replication is 1,000.

The first column shows the possible values that σ^{(s _t )} can take. Notice that in the regime-specific case, σ^{(s _t )} is set to 0.5 if s _t  = regime 1 and 1 if s _t  = regime 2. The second column shows the relevant information criterion being considered, while the third column indicates the number of regimes imposed on estimation. With sample size of 100, the fourth column depicts the frequency of selecting the correct number of variable (in this case, 1 variable) while the fifth column indicates the frequency of selecting the incorrect number of variables (in this case, 2 variables). Other columns show similar frequencies for different sample sizes.

The true data generating process is the MS static model with two regimes and one exogenous variable, as follows:

The initial probabilities are set to 0.4 and 0.6 with the transition probabilities as p ₁₁ = 0.6; p ₂₂ = 0.4. The true regression coefficients are v ^{(s _t )} = (v ⁽¹⁾, v ⁽²⁾) and A ^{(s _t )} = (A ⁽¹⁾, A ⁽²⁾), where v ^{(s _t )} = (0, 1) and A ^{(s _t )} = (0.3, 0.9). Notice that the total replication is 1,000.

The first column shows the possible values that σ^{(s _t )} can take. Notice that in the regime-specific case, σ^{(s _t )} is set to 0.5 if s _t  = regime 1 and 1 if s _t  = regime 2. The second column shows the relevant information criterion being considered, while the third column indicates the number of regimes imposed on estimation. With sample size of 100, the fourth column depicts the frequency of selecting the correct number of variable (in this case, 1 variable) while the fifth column indicates the frequency of selecting the incorrect number of variables (in this case, 2 variables). Other columns show similar frequencies for different sample sizes.

The true data generating process is the MS static model with two regimes and one exogenous variable, as follows:

The initial probabilities are set to 0.4 and 0.6 with the transition probabilities as p ₁₁ = 0.9; p ₂₂ = 0.1. The true regression coefficients are v ^{(s _t )} = (v ⁽¹⁾, v ⁽²⁾) and A ^{(s _t )} = (A ⁽¹⁾, A ⁽²⁾), where v ^{(s _t )} = (0, 0) and A ^{(s _t )} = (0.3, 0.9). Notice that the total replication is 1,000.

The first column shows the possible values that σ^{(s _t )} can take. Notice that in the regime-specific case, σ^{(s _t )} is set to 0.5 if s _t  = regime 1 and 1 if s _t  = regime 2. The second column shows the relevant information criterion being considered, while the third column indicates the number of regimes imposed on estimation. With sample size of 100, the fourth column depicts the frequency of selecting the correct number of variable (in this case, 1 variable) while the fifth column indicates the frequency of selecting the incorrect number of variables (in this case, 2 variables). Other columns show similar frequencies for different sample sizes.

The true data generating process is the MS static model with two regimes and one exogenous variable, as follows:

The initial probabilities are set to 0.5 and 0.5 with the transition probabilities as p ₁₁ = 0.6; p ₂₂ = 0.4. The true regression coefficients are v ^{(s _t )} = (v ⁽¹⁾, v ⁽²⁾) and A ^{(s _t )} = (A ⁽¹⁾, A ⁽²⁾), where v ^{(s _t )} = (0, 0) and A ^{(s _t )} = (0.3, 0.9). Notice that the total replication is 1,000.

The first column shows the possible values that σ^{(s _t )} can take. Notice that in the regime-specific case, σ^{(s _t )} is set to 0.5 if s _t  = regime 1 and 1 if s _t  = regime 2. The second column shows the relevant information criterion being considered, while the third column indicates the number of regimes imposed on estimation. With sample size of 100, the fourth column depicts the frequency of selecting the correct number of variable (in this case, 1 variable) while the fifth column indicates the frequency of selecting the incorrect number of variables (in this case, 2 variables). Other columns show similar frequencies for different sample sizes.

The true data generating process is the MS static model with two regimes and one exogenous variable, as follows:

The initial probabilities are set to 0.4 and 0.6 with the transition probabilities as p ₁₁ = 0.9; p ₂₂ = 0.9. The true regression coefficients are v ^{(s _t )} = (v ⁽¹⁾, v ⁽²⁾) and A ^{(s _t )} = (A ⁽¹⁾, A ⁽²⁾), where v ^{(s _t )} = (0, 0) and A ^{(s _t )} = (0.3, 0.9). Notice that the total replication is 1,000.

The first column shows the possible values that σ^{(s _t )} can take. Notice that in the regime-specific case, σ^{(s _t )} is set to 0.5 if s _t  = regime 1 and 1 if s _t  = regime 2. The second column shows the relevant information criterion being considered, while the third column indicates the number of regimes imposed on estimation. With sample size of 100, the fourth column depicts the frequency of selecting the correct number of variable (in this case, 1 variable) while the fifth column indicates the frequency of selecting the incorrect number of variables (in this case, 2 variables). Other columns show similar frequencies for different sample sizes.

The true data generating process is the MS-AR model with two regimes and one lag, as follows:

The initial probabilities are set to 0.4 and 0.6 with the transition probabilities as p ₁₁ = 0.6; p ₂₂ = 0.4. The true regression coefficients are v ^{(s _t )} = (v ⁽¹⁾, v ⁽²⁾) and A ^{(s _t )} = (A ⁽¹⁾, A ⁽²⁾), where v ^{(s _t )} = (0, 0) and A ^{(s _t )} = (0.3, 0.9). Notice that the total replication is 1,000.

The first column shows the possible values that σ^{(s _t )} can take. Notice that in the regime-specific case, σ^{(s _t )} is set to 0.5 if s _t  = regime 1 and 1 if s _t  = regime 2. The second column shows the relevant information criterion being considered, while the third column indicates the number of regimes imposed on estimation. With sample size of 100, the fourth column depicts the frequency of selecting the correct lag length (in this case, 1 lag) while the fifth and sixth columns indicate the frequency of selecting the incorrect lag lengths (in this case, 2 and 3 lags). Other columns show similar frequencies for different sample sizes.

The true data generating process is the MS-AR model with two regimes and one lag, as follows:

The initial probabilities are set to 0.4 and 0.6 with the transition probabilities as p ₁₁ = 0.6; p ₂₂ = 0.4. The true regression coefficients are v ^{(s _t )} = (v ⁽¹⁾, v ⁽²⁾) and A ^{(s _t )} = (A ⁽¹⁾, A ⁽²⁾), where v ^{(s _t )} = (0, 1) and A ^{(s _t )} = (0.3, 0.9). Notice that the total replication is 1,000.

The first column shows the possible values that σ^{(s _t )} can take. Notice that in the regime-specific case, σ^{(s _t )} is set to 0.5 if s _t  = regime 1 and 1 if s _t  = regime 2. The second column shows the relevant information criterion being considered, while the third column indicates the number of regimes imposed on estimation. With sample size of 100, the fourth column depicts the frequency of selecting the correct lag length (in this case, 1 lag) while the fifth and sixth columns indicate the frequency of selecting the incorrect lag lengths (in this case, 2 and 3 lags). Other columns show similar frequencies for different sample sizes.

The true data generating process is the MS-AR model with two regimes and one lag, as follows:

The initial probabilities are set to 0.4 and 0.6 with the transition probabilities as p ₁₁ = 0.9; p ₂₂ = 0.1. The true regression coefficients are v ^{(s _t )} = (v ⁽¹⁾, v ⁽²⁾) and A ^{(s _t )} = (A ⁽¹⁾, A ⁽²⁾), where v ^{(s _t )} = (0, 0) and A ^{(s _t )} = (0.3, 0.9). Notice that the total replication is 1,000.

The first column shows the possible values that σ^{(s _t )} can take. Notice that in the regime-specific case, σ^{(s _t )} is set to 0.5 if s _t  = regime 1 and 1 if s _t  = regime 2. The second column shows the relevant information criterion being considered, while the third column indicates the number of regimes imposed on estimation. With sample size of 100, the fourth column depicts the frequency of selecting the correct lag length (in this case, 1 lag) while the fifth and sixth columns indicate the frequency of selecting the incorrect lag lengths (in this case, 2 and 3 lags). Other columns show similar frequencies for different sample sizes.

The true data generating process is the MS-AR model with two regimes and one lag, as follows:

The initial probabilities are set to 0.1 and 0.9 with the transition probabilities as p ₁₁ = 0.6; p ₂₂ = 0.4. The true regression coefficients are v ^{(s _t )} = (v ⁽¹⁾, v ⁽²⁾) and A ^{(s _t )} = (A ⁽¹⁾, A ⁽²⁾), where v ^{(s _t )} = (0, 0) and A ^{(s _t )} = (0.3, 0.9). Notice that the total replication is 1,000.

The first column shows the possible values that σ^{(s _t )} can take. Notice that in the regime-specific case, σ^{(s _t )} is set to 0.5 if s _t  = regime 1 and 1 if s _t  = regime 2. The second column shows the relevant information criterion being considered, while the third column indicates the number of regimes imposed on estimation. With sample size of 100, the fourth column depicts the frequency of selecting the correct lag length (in this case, 1 lag) while the fifth and sixth columns indicate the frequency of selecting the incorrect lag lengths (in this case, 2 and 3 lags). Other columns show similar frequencies for different sample sizes.

The true data generating process is the MS-AR model with two regimes and one lag, as follows:

The initial probabilities are set to 0.4 and 0.6 with the transition probabilities as p ₁₁ = 0.9; p ₂₂ = 0.9. The true regression coefficients are v ^{(s _t )} = (v ⁽¹⁾, v ⁽²⁾) and A ^{(s _t )} = (A ⁽¹⁾, A ⁽²⁾), where v ^{(s _t )} = (0, 0) and A ^{(s _t )} = (0.3, 0.9). Notice that the total replication is 1,000.

The first column shows the possible values that σ^{(s _t )} can take. Notice that in the regime-specific case, σ^{(s _t )} is set to 0.5 if s _t  = regime 1 and 1 if s _t  = regime 2. The second column shows the relevant information criterion being considered, while the third column indicates the number of regimes imposed on estimation. With sample size of 100, the fourth column depicts the frequency of selecting the correct lag length (in this case, 1 lag) while the fifth and sixth columns indicate the frequency of selecting the incorrect lag lengths (in this case, 2 and 3 lags). Other columns show similar frequencies for different sample sizes.