# Selecting a Model for Forecasting

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## Abstract

**:**

## 1. Introduction

## 2. Empirical Motivation

## 3. The Analytic Design

## 4. Selection in a Stationary DGP

#### 4.1. Known Future Values of Regressors

#### 4.2. Selecting Regressors

#### 4.3. The Choice of Significance Level

## 5. An Out-of-Sample Shift in the Regressors

#### 5.1. Specification of the Out-of-Sample Shift

#### 5.2. Known Future Values of Regressors

#### 5.3. Selecting Regressors

#### 5.4. Unknown Future Values of Regressors

#### 5.5. Forecasting Regressors with a Random Walk

#### 5.6. Selecting Forecasted Regressors

## 6. An In-Sample Shift in the Regressors

#### 6.1. Specification of the In-Sample Shift

#### 6.2. Forecasting Regressors Using In-Sample Means

#### 6.3. Selecting Regressors

#### 6.4. Forecasting Regressors Using a Random Walk

#### 6.5. Selecting Forecasted Regressors

#### 6.6. Forecasting the Dependent Variable Using a Random Walk

## 7. Summary of Analytic Results and the Impact of Selection

- Regressors should be retained if ${\psi}_{\beta}\ge 1$. This is established for DGPs that are stationary or with a break out of sample for known regressors and a break in sample for random walk forecasts.
- For the two-regressor case, ${\psi}_{\beta}=1$ maps to $\alpha \approx 0.16$. Selection delivers improvements to the one-step-ahead MSFE for ${\psi}_{\beta}<1$ and can be close to the correct model specification for ${\psi}_{\beta}>1$, with the largest deviations occurring at intermediate values of ${\psi}_{\beta}$.
- If there are breaks out of sample and contemporaneous regressors need to be forecast, the break dominates the MSFE and selection plays almost no role. Similar results are found even if the break occurs at the end of the sample, but the in-sample mean is used to forecast the regressors.
- Random walk forecasts are costly if there are no breaks (forecasting ${x}_{1,T+1}$) or if the breaks are unpredictable (a break at $T+1$ and forecasting $T+1|T$). However, they improve MSFE when the break is predictable (break at T and forecasting $T+1|T$).

## 8. Simulation Design

#### 8.1. Data Generation Process

#### 8.2. Models and Forecast Devices

**inf:**- future outcomes: ${\tilde{x}}_{j,T+h}={x}_{j,T+h}$;
**avg:**- the in-sample average: ${\tilde{x}}_{j,T+h}={\sum}_{t=h}^{T+h-1}{x}_{j,t}/T$;
**arx:**- an AR(1) for each regressor: ${\tilde{x}}_{j,T+h}={\widehat{\mu}}_{j}+{\widehat{\rho}}_{j}{x}_{j,T+h-1}$, estimated by OLS for each horizon from:$${x}_{j,t}={\mu}_{j}+{\rho}_{j}{x}_{j,t-1}+{u}_{j,t},\phantom{\rule{1.em}{0ex}}t=h,\dots ,T+h-1;$$
**rwx:**- the random walk forecast: ${\tilde{x}}_{j,T+h}={x}_{j,T+h-1}$;
**rdx:**- a random walk with differencing (Hendry 2006), using differenced estimates from (39):$${\tilde{x}}_{j,T+h}={x}_{j,T+h-1}+{\widehat{\rho}}_{j}\Delta {x}_{j,T+h-1}.$$
**cax:**- Cardt forecast of ${\tilde{x}}_{j,T+h}$.

**rwy:**- a random walk forecast: ${\widehat{y}}_{T+h}={y}_{T+h-1}$;
**ary:**- an AR(1) forecast: ${\widehat{y}}_{T+h}={\widehat{\gamma}}_{0}+{\widehat{\gamma}}_{1}{y}_{T+h-1}$, estimated by OLS for each horizon;
**cay:**- Cardt forecasts of ${\widehat{y}}_{T+h}$.

#### 8.3. Selecting Regressors

## 9. Simulation Evidence

#### 9.1. Forecasting before the Break

#### 9.2. Selection and Location of the Break

**Break in relevant regressors**- (${\delta}_{R}=-0.3,{\lambda}_{R}=0.05,{\delta}_{I}=\delta ,{\lambda}_{I}=\lambda $)The break shows up in y through the relevant variables. Inclusion of irrelevant variables in the forecasting model is not costly relative to the impact of the break. Loose selection is preferred, because it includes more relevant variables. For $T+1|T$ selection has no impact because the break is not observed (except for known regressors). Including regressors in arx and rwx gives a substantial improvement over ary.
**Break in irrelevant regressors**- (${\delta}_{I}=-0.3,{\lambda}_{I}=0.05,{\delta}_{R}=\delta ,{\lambda}_{R}=\lambda $)There is no break in y, so any inclusion of irrelevant variables is costly, as their break offsets the small estimated coefficients. The more irrelevant variables included, the stronger this effect. The autoregression in y is almost always preferred.
**Break in all regressors**- (${\delta}_{R}={\delta}_{I}=-0.3,{\lambda}_{R}={\lambda}_{I}=0.05$)The y variable is identical to that of a break in relevant variables only. Selection is now a trade-off between including variables that matter and help with forecasting, and irrelevant variables that make forecasts worse. Including regressors in arx and rwx gives a substantial improvement over ary.

#### 9.3. Forecasting after theBreak

#### 9.4. Is Selection Costly When Forecasting?

#### 9.5. Forecast Combinations

**apool**- (arx + rwy)/2;
**cpool**- (arx + cay)/2.

#### 9.6. Summary of the Simulation Results

## 10. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Analytic Calculations

#### Appendix A.1.

#### Appendix A.2.

#### Appendix A.3.

#### Appendix A.4.

#### Appendix A.5.

#### Appendix A.6.

#### Appendix A.7.

## Appendix B

MSFE Relative to ${\mathbf{MSFE}}_{1}$ | |||||
---|---|---|---|---|---|

Model | ${\mathit{\psi}}_{\mathit{\beta}}^{2}=0$ | ${\mathit{\psi}}_{\mathit{\beta}}^{2}=1$ | ${\mathit{\psi}}_{\mathit{\beta}}^{2}=4$ | ${\mathit{\psi}}_{\mathit{\beta}}^{2}=9$ | ${\mathit{\psi}}_{\mathit{\beta}}^{2}=16$ |

Section 4.1 and Section 4.2 No shift with known future regressors | |||||

$\alpha =0\phantom{\rule{4pt}{0ex}}\left({\mathrm{M}}_{2}\right)$ | 0.990 | 1.000 | 1.030 | 1.079 | 1.149 |

$\alpha =0.001$ | 0.990 | 1.000 | 1.026 | 1.048 | 1.035 |

$\alpha =0.05$ | 0.991 | 1.000 | 1.014 | 1.012 | 1.003 |

$\alpha =0.16$ | 0.992 | 1.000 | 1.008 | 1.004 | 1.001 |

Section 5.2 and Section 5.3 Out-of-sample shift with known future regressors | |||||

$\alpha =0\phantom{\rule{4pt}{0ex}}\left({\mathrm{M}}_{2}\right)$ | 0.827 | 1.008 | 1.551 | 2.457 | 3.724 |

$\alpha =0.001$ | 0.827 | 1.008 | 1.497 | 1.895 | 1.651 |

$\alpha =0.05$ | 0.836 | 1.007 | 1.267 | 1.217 | 1.056 |

$\alpha =0.16$ | 0.855 | 1.005 | 1.152 | 1.081 | 1.013 |

Section 5.4 Out-of-sample shift with mean forecast of future regressors | |||||

$\alpha =0\phantom{\rule{4pt}{0ex}}\left({\mathrm{M}}_{2}\right)$ | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

$\alpha =0.001$ | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

$\alpha =0.05$ | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

$\alpha =0.16$ | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

Section 5.5 Out-of-sample shift with random walk forecast of future regressors | |||||

$\alpha =0\phantom{\rule{4pt}{0ex}}\left({\mathrm{M}}_{2}\right)$ | 0.997 | 1.002 | 1.013 | 1.024 | 1.033 |

$\alpha =0.001$ | 0.997 | 1.002 | 1.012 | 1.015 | 1.008 |

$\alpha =0.05$ | 0.997 | 1.002 | 1.006 | 1.004 | 1.001 |

$\alpha =0.16$ | 0.997 | 1.001 | 1.004 | 1.001 | 1.000 |

Section 6.2 and Section 6.3 In-sample shift with mean forecast of future regressors | |||||

$\alpha =0\phantom{\rule{4pt}{0ex}}\left({\mathrm{M}}_{2}\right)$ | 1.010 | 1.009 | 1.008 | 1.007 | 1.007 |

$\alpha =0.001$ | 1.010 | 1.009 | 1.008 | 1.005 | 1.002 |

$\alpha =0.05$ | 1.010 | 1.008 | 1.004 | 1.001 | 1.000 |

$\alpha =0.16$ | 1.008 | 1.006 | 1.002 | 1.000 | 1.000 |

Section 6.4 and Section 6.5 In-sample shift with random walk forecast of future regressors | |||||

$\alpha =0\phantom{\rule{4pt}{0ex}}\left({\mathrm{M}}_{2}\right)$ | 0.931 | 0.994 | 1.155 | 1.386 | 1.661 |

$\alpha =0.001$ | 0.931 | 0.994 | 1.140 | 1.237 | 1.158 |

$\alpha =0.05$ | 0.934 | 0.995 | 1.075 | 1.058 | 1.014 |

$\alpha =0.16$ | 0.942 | 0.996 | 1.043 | 1.021 | 1.003 |

## Note

1 | Clements and Hendry (1993) argue that the generalized forecast error second moment should be used to evaluate forecast performance instead of MSFE. In this case the results would be equivalent, because we focus on one-step-ahead forecasts. |

2 | UK quarterly consumer price index (CPI) is given by ONS series D7BT, which is the quarterly average of the monthly index. Annual inflation percentage is defined as ${\pi}_{t}=100{\Delta}_{4}log{\mathrm{D}7\mathrm{BT}}_{t}$. UK Unemployment is the quarterly average of ONS series MGUK, LFS ILO unemployment rate (UK, All, Aged 16 and over, %, NSA). |

3 | Intermediate alternatives such as sub-sample estimation, recursive or rolling estimation could also be used. |

4 | Castle et al. (2012) demonstrate the ability of IIS to detect breaks in the form of location shifts at any point in the sample. |

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**Figure 1.**(

**a**) Quarterly average of CPI 12 month inflation rates for the UK (percent per annum); (

**b**) quarterly UK unemployment rate in percent, with SIS detected mean shifts at $\alpha =0.1\%$.

**Figure 3.**The costs/benefits of selection measured by $\frac{{\mathrm{MSFE}}_{3}}{{\mathrm{MSFE}}_{1}}$ in (14).

**Figure 4.**Values of $(1-{\mathrm{p}}_{\alpha}\left({\psi}_{\beta}\right))$ for five independent regressors with the same noncentrality for a range of $\alpha $ and ${\psi}_{\beta}^{2}$.

**Figure 5.**MSFE comparisons of ${\mathrm{M}}_{1}$, ${\mathrm{M}}_{2}$ and ${\mathrm{M}}_{3}$ at 3 illustrative values of $\alpha $ for known future exogenous regressors where the break occurs in the mean of ${x}_{2}$ at $T+1$.

**Figure 6.**MSFE comparisons between ${\mathrm{M}}_{1}$, ${\mathrm{M}}_{2}$ and ${\mathrm{M}}_{3}$ for known and unknown future exogenous regressors including in-sample mean and random walk forecasts, where the break occurs in the mean of ${x}_{2}$ at $T+1$.

**Figure 7.**${\mathrm{MSFE}}_{1}$, ${\mathrm{MSFE}}_{2}$, and ${\mathrm{MSFE}}_{3}$ for unknown future exogenous regressors where the break occurs in the mean of ${x}_{2}$ at T and the in-sample mean is used as the forecast for the regressors. Included are the results when the break occurs at $T+1$.

**Figure 8.**MSFE comparisons between ${\mathrm{M}}_{1}$, ${\mathrm{M}}_{2}$ and ${\mathrm{M}}_{3}$ at $\alpha =0.16$ for unknown future exogenous regressors where the break occurs in the mean of ${x}_{2}$ at T and the last in-sample observation is used as the forecast for the conditioning regressors. Also recorded is the MSFE for ${\mathrm{M}}_{1}$ and ${\mathrm{M}}_{2}$ using in-sample means and a misspecified random walk for ${y}_{T+1}$ directly.

**Figure 9.**One replication of the DGP without break (solid line) and breaks as in Table 5, $T=100,H=5$.

**Table 1.**Root mean square error of one-step forecast for $\Delta {\pi}_{t}$ over the period 2014Q1–2017Q4.

Conditioning on | M_{1} | M_{2} | M_{3} |
---|---|---|---|

Known ${U}_{t}$ | 0.535 | 0.530 | 0.515 |

Mean forecast for ${U}_{t}$ | 0.519 | 0.530 | 0.542 |

Random walk forecast for ${U}_{t}$ | 0.549 | 0.530 | 0.515 |

**Table 2.**Retention probabilities for individual t-tests given $\mathrm{E}\left[{\mathrm{t}}_{{\widehat{\beta}}_{2}}\right]={\psi}_{\beta}$.

${\mathit{\psi}}_{\mathit{\beta}}$ | 1 | 2 | 3 | 4 |
---|---|---|---|---|

${\mathrm{P}}_{0.16}$ | 0.34 | 0.72 | 0.94 | 0.995 |

${\mathrm{P}}_{0.05}$ | 0.16 | 0.51 | 0.85 | 0.98 |

**Table 3.**Ratio of MSFE to that of ${\mathrm{MSFE}}_{1}$, $T=50$. ${\mathrm{M}}_{2}$ has no selection ($\alpha =0$); selection in ${\mathrm{M}}_{3}$ at $\alpha $.

MSFE Relative to ${\mathbf{MSFE}}_{1}$ | ||||||
---|---|---|---|---|---|---|

Model | ${\mathit{\psi}}_{\mathit{\beta}}^{2}=0$ | ${\mathit{\psi}}_{\mathit{\beta}}^{2}=1$ | ${\mathit{\psi}}_{\mathit{\beta}}^{2}=4$ | ${\mathit{\psi}}_{\mathit{\beta}}^{2}=9$ | ${\mathit{\psi}}_{\mathit{\beta}}^{2}=16$ | |

Section 4.1 and Section 4.2 No shift with known future regressors | ||||||

$\alpha =0\phantom{\rule{4pt}{0ex}}\left({\mathrm{M}}_{2}\right)$ | 0.981 | 1.001 | 1.060 | 1.158 | 1.295 | |

$\alpha =0.001$ | 0.981 | 1.000 | 1.051 | 1.093 | 1.068 | |

$\alpha =0.05$ | 0.982 | 1.000 | 1.027 | 1.023 | 1.006 | |

$\alpha =0.16$ | 0.984 | 1.000 | 1.016 | 1.008 | 1.001 | |

Section 5.2 and Section 5.3 Out-of-sample shift with known future regressors | ||||||

$\alpha =0\phantom{\rule{4pt}{0ex}}\left({\mathrm{M}}_{2}\right)$ | 0.709 | 1.014 | 1.927 | 3.450 | 5.582 | |

$\alpha =0.001$ | 0.709 | 1.013 | 1.836 | 2.505 | 2.095 | |

$\alpha =0.05$ | 0.724 | 1.011 | 1.449 | 1.366 | 1.095 | |

$\alpha =0.16$ | 0.756 | 1.009 | 1.256 | 1.136 | 1.022 | |

Section 5.4 Out-of-sample shift with mean forecast of future regressors | ||||||

$\alpha =0\phantom{\rule{4pt}{0ex}}\left({\mathrm{M}}_{2}\right)$ | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | |

$\alpha =0.001$ | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | |

$\alpha =0.05$ | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | |

$\alpha =0.16$ | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | |

Section 5.5 Out-of-sample shift with random walk forecast of future regressors | ||||||

$\alpha =0\phantom{\rule{4pt}{0ex}}\left({\mathrm{M}}_{2}\right)$ | 0.993 | 1.004 | 1.020 | 1.034 | 1.043 | |

$\alpha =0.001$ | 0.993 | 1.004 | 1.018 | 1.021 | 1.010 | |

$\alpha =0.05$ | 0.994 | 1.003 | 1.010 | 1.005 | 1.001 | |

$\alpha =0.16$ | 0.994 | 1.002 | 1.006 | 1.002 | 1.000 | |

Section 6.2 and Section 6.3 In-sample shift with mean forecast of future regressors | ||||||

$\alpha =0\phantom{\rule{4pt}{0ex}}\left({\mathrm{M}}_{2}\right)$ | 1.020 | 1.021 | 1.022 | 1.023 | 1.024 | |

$\alpha =0.001$ | 1.020 | 1.021 | 1.020 | 1.014 | 1.006 | |

$\alpha =0.05$ | 1.019 | 1.017 | 1.011 | 1.004 | 1.000 | |

$\alpha =0.16$ | 1.017 | 1.014 | 1.006 | 1.001 | 1.000 | |

Section 6.4 and Section 6.5 In-sample shift with random walk forecast of future regressors | ||||||

$\alpha =0\phantom{\rule{4pt}{0ex}}\left({\mathrm{M}}_{2}\right)$ | 0.871 | 0.990 | 1.273 | 1.653 | 2.078 | |

$\alpha =0.001$ | 0.871 | 0.990 | 1.246 | 1.401 | 1.258 | |

$\alpha =0.05$ | 0.878 | 0.991 | 1.132 | 1.097 | 1.022 | |

$\alpha =0.16$ | 0.892 | 0.993 | 1.075 | 1.036 | 1.005 |

**Table 4.**Impact on x when coefficients change from $(\delta ,\lambda )=(2,0.75)$ to $({\delta}_{\Delta},{\lambda}_{\Delta})$.

$({\mathit{\delta}}_{\mathsf{\Delta}},{\mathit{\lambda}}_{\mathsf{\Delta}})=$ | (2, 0.75) | (−0.3, 0.75) | (−0.3, 0.95) | (−0.3, 0.05) | (2, 0.05) | (2, 0.95) |
---|---|---|---|---|---|---|

${x}_{j,T+1|T}$ | 8 | 5.7 | 7.3 | 0.1 | 2.4 | 9.6 |

${x}_{j,T+2|T+1}$ | 8 | 4.0 | 6.6 | −0.3 | 2.1 | 11.1 |

${x}_{j,T+3|T+2}$ | 8 | 5.0 | 7.0 | 1.8 | 3.6 | 10.3 |

${\mathit{\delta}}_{\mathsf{\Delta}}$ | ${\mathit{\lambda}}_{\mathsf{\Delta}}$ | |
---|---|---|

No break | 2 | 0.75 |

Break in mean | −0.3 | 0.75 |

Break in slope (a) | 2 | 0.95 |

Break in slope (b) | 2 | 0.05 |

Break in mean and slope (a) | −0.3 | 0.95 |

Break in mean and slope (b) | −0.3 | 0.05 |

**Table 6.**Probability of retaining one or all variables when the coefficients have the specified noncentrality, assuming independence at nominal significance $\alpha $ and Student-t(83) distribution.

${\mathit{\psi}}_{\mathit{\beta}}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}1.2$ | ${\mathit{\psi}}_{\mathit{\beta}}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.5$ | ${\mathit{\psi}}_{\mathit{\beta}}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}1$ | ${\mathit{\psi}}_{\mathit{\beta}}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}1.5$ | ${\mathit{\psi}}_{\mathit{\beta}}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}2$ | ${\mathit{\psi}}_{\mathit{\beta}}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}3$ | ${\mathit{\psi}}_{\mathit{\beta}}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}4$ | Joint | Average | ${\mathit{\psi}}_{\mathit{\beta}}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}4$ | |||
---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{\alpha}$ | $\mathit{n}\phantom{\rule{3.33333pt}{0ex}}\mathbf{=}\phantom{\rule{3.33333pt}{0ex}}\mathbf{1}$ | $\mathit{n}\phantom{\rule{3.33333pt}{0ex}}\mathbf{=}\phantom{\rule{3.33333pt}{0ex}}\mathbf{10}$ | $\mathit{n}\phantom{\rule{3.33333pt}{0ex}}\mathbf{=}\phantom{\rule{3.33333pt}{0ex}}\mathbf{1}$ | $\mathit{n}\phantom{\rule{3.33333pt}{0ex}}\mathbf{=}\phantom{\rule{3.33333pt}{0ex}}\mathbf{1}$ | $\mathit{n}\phantom{\rule{3.33333pt}{0ex}}\mathbf{=}\phantom{\rule{3.33333pt}{0ex}}\mathbf{1}$ | $\mathit{n}\phantom{\rule{3.33333pt}{0ex}}\mathbf{=}\phantom{\rule{3.33333pt}{0ex}}\mathbf{1}$ | $\mathit{n}\phantom{\rule{3.33333pt}{0ex}}\mathbf{=}\phantom{\rule{3.33333pt}{0ex}}\mathbf{1}$ | $\mathit{n}\phantom{\rule{3.33333pt}{0ex}}\mathbf{=}\phantom{\rule{3.33333pt}{0ex}}\mathbf{1}$ | $\mathit{n}\phantom{\rule{3.33333pt}{0ex}}\mathbf{=}\phantom{\rule{3.33333pt}{0ex}}\mathbf{6}$ | $\mathit{n}\phantom{\rule{3.33333pt}{0ex}}\mathbf{=}\phantom{\rule{3.33333pt}{0ex}}\mathbf{6}$ | $\mathit{n}\phantom{\rule{3.33333pt}{0ex}}\mathbf{=}\phantom{\rule{3.33333pt}{0ex}}\mathbf{1}$ | $\mathit{n}\phantom{\rule{3.33333pt}{0ex}}\mathbf{=}\phantom{\rule{3.33333pt}{0ex}}\mathbf{3}$ |

0.001 | 0.015 | 0.000 | 0.002 | 0.009 | 0.030 | 0.081 | 0.341 | 0.721 | 0.000 | 0.197 | 0.721 | 0.375 |

0.01 | 0.077 | 0.000 | 0.018 | 0.053 | 0.130 | 0.263 | 0.641 | 0.912 | 0.000 | 0.336 | 0.912 | 0.758 |

0.05 | 0.216 | 0.000 | 0.070 | 0.163 | 0.313 | 0.504 | 0.843 | 0.976 | 0.001 | 0.478 | 0.976 | 0.930 |

0.1 | 0.322 | 0.000 | 0.124 | 0.254 | 0.435 | 0.631 | 0.907 | 0.989 | 0.008 | 0.557 | 0.989 | 0.968 |

0.16 | 0.414 | 0.000 | 0.181 | 0.339 | 0.533 | 0.719 | 0.941 | 0.994 | 0.022 | 0.618 | 0.994 | 0.983 |

0.32 | 0.579 | 0.004 | 0.309 | 0.500 | 0.691 | 0.840 | 0.976 | 0.998 | 0.087 | 0.719 | 0.998 | 0.995 |

Gauge | Potency | |||||
---|---|---|---|---|---|---|

$\mathit{\alpha}$ | $\mathit{\psi}\mathbf{\left(}\mathbf{1}\mathbf{\right)}$ | $\mathit{\psi}\mathbf{\left(}\mathbf{2}\mathbf{\right)}$ | $\mathit{\psi}\mathbf{\left(}\mathbf{4}\mathbf{\right)}$ | $\mathit{\psi}\mathbf{\left(}\mathbf{1}\mathbf{\right)}$ | $\mathit{\psi}\mathbf{\left(}\mathbf{2}\mathbf{\right)}$ | $\mathit{\psi}\mathbf{\left(}\mathbf{4}\mathbf{\right)}$ |

0.001 | 0.005 | 0.006 | 0.006 | 0.034 | 0.205 | 0.712 |

0.01 | 0.025 | 0.024 | 0.020 | 0.113 | 0.345 | 0.884 |

0.05 | 0.079 | 0.075 | 0.069 | 0.231 | 0.458 | 0.919 |

0.1 | 0.126 | 0.124 | 0.121 | 0.297 | 0.507 | 0.919 |

0.16 | 0.181 | 0.180 | 0.178 | 0.355 | 0.545 | 0.923 |

0.32 | 0.328 | 0.328 | 0.327 | 0.479 | 0.634 | 0.941 |

$\mathit{\psi}\left(1\right)$ | $\mathit{\psi}\left(2\right)$ | $\mathit{\psi}\left(4\right)$ | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

inf | avg | arx | rwx | inf | avg | arx | rwx | inf | avg | arx | rwx | |

Ratio | No break | |||||||||||

$\alpha \phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0$ | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |

$\alpha \phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.001$ | 1.03 | 1.01 | 1.02 | 1.03 | 0.95 | 1.06 | 0.99 | 1.02 | 0.83 | 1.13 | 0.94 | 0.99 |

$\alpha \phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.01$ | 1.08 | 1.06 | 1.05 | 1.08 | 0.93 | 1.11 | 0.98 | 1.02 | 0.79 | 1.17 | 0.93 | 0.97 |

$\alpha \phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.05$ | 1.13 | 1.13 | 1.08 | 1.12 | 0.95 | 1.19 | 0.99 | 1.03 | 0.83 | 1.23 | 0.95 | 0.99 |

$\alpha \phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.1$ | 1.16 | 1.18 | 1.09 | 1.13 | 0.99 | 1.23 | 1.01 | 1.06 | 0.87 | 1.27 | 0.97 | 1.02 |

$\alpha \phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.16$ | 1.19 | 1.23 | 1.11 | 1.15 | 1.01 | 1.28 | 1.04 | 1.08 | 0.91 | 1.31 | 1.00 | 1.04 |

$\alpha \phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.32$ | 1.25 | 1.36 | 1.15 | 1.19 | 1.09 | 1.38 | 1.09 | 1.13 | 0.99 | 1.41 | 1.05 | 1.09 |

GUM | 1.34 | 1.51 | 1.20 | 1.23 | 1.18 | 1.50 | 1.13 | 1.17 | 1.08 | 1.52 | 1.10 | 1.14 |

MSFEary | 1.15 | 1.31 | 1.43 | |||||||||

Ratio | Average over five break types in relevant regressors | |||||||||||

$\alpha \phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0$ | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |

$\alpha \phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.001$ | 0.90 | 1.00 | 1.00 | 1.01 | 0.58 | 1.02 | 0.99 | 1.00 | 0.37 | 1.05 | 0.97 | 0.98 |

$\alpha \phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.01$ | 0.74 | 1.01 | 1.01 | 1.02 | 0.42 | 1.04 | 0.99 | 0.99 | 0.28 | 1.06 | 0.97 | 0.97 |

$\alpha \phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.05$ | 0.57 | 1.03 | 1.01 | 1.02 | 0.37 | 1.06 | 0.99 | 0.99 | 0.28 | 1.08 | 0.97 | 0.98 |

$\alpha \phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.1$ | 0.52 | 1.05 | 1.02 | 1.03 | 0.37 | 1.07 | 0.99 | 1.00 | 0.29 | 1.09 | 0.98 | 0.98 |

$\alpha \phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.16$ | 0.50 | 1.06 | 1.02 | 1.03 | 0.37 | 1.08 | 1.00 | 1.01 | 0.30 | 1.10 | 0.99 | 0.99 |

$\alpha \phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.32$ | 0.48 | 1.10 | 1.03 | 1.04 | 0.38 | 1.11 | 1.01 | 1.02 | 0.32 | 1.13 | 1.00 | 1.01 |

GUM | 0.49 | 1.13 | 1.05 | 1.05 | 0.41 | 1.14 | 1.03 | 1.03 | 0.35 | 1.16 | 1.01 | 1.02 |

MSFEary | 18.58 | 18.80 | 18.98 |

$\mathit{T}+1|\mathit{T}$ | $\mathit{T}+2|\mathit{T}+1$ | $\mathit{T}+3|\mathit{T}+2$ | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{\psi}$ | Where | inf | avg | arx | rwx | inf | avg | arx | rwx | inf | avg | arx | rwx | |

$\alpha \phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0$ary | $\mathit{\psi}\left(1\right)$ | Relevant | 54.42 | 50.41 | 6.75 | |||||||||

$\alpha \phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.1$ | $\mathit{\psi}\left(1\right)$ | Relevant | 16.44 | 54.49 | 54.50 | 54.60 | 10.67 | 60.24 | 18.33 | 11.24 | 3.51 | 33.30 | 3.03 | 3.48 |

GUM | $\mathit{\psi}\left(1\right)$ | Relevant | 11.43 | 54.78 | 54.63 | 54.77 | 10.15 | 60.60 | 13.56 | 9.76 | 2.89 | 41.42 | 2.43 | 4.15 |

$\alpha \phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0$ary | $\mathit{\psi}\left(1\right)$ | All | 54.42 | 50.41 | 6.75 | |||||||||

$\alpha \phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.1$ | $\mathit{\psi}\left(1\right)$ | All | 18.32 | 54.49 | 54.50 | 54.60 | 11.19 | 61.32 | 18.71 | 11.70 | 3.32 | 33.28 | 2.89 | 3.61 |

GUM | $\mathit{\psi}\left(1\right)$ | All | 16.42 | 54.78 | 54.63 | 54.77 | 14.12 | 64.35 | 17.41 | 13.64 | 3.05 | 42.12 | 2.55 | 4.21 |

$\alpha \phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0$ary | $\mathit{\psi}\left(1\right)$ | Irrel. | 1.15 | 1.19 | 1.18 | |||||||||

$\alpha \phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.1$ | $\mathit{\psi}\left(1\right)$ | Irrel. | 3.19 | 1.36 | 1.26 | 1.31 | 2.86 | 2.59 | 2.55 | 2.80 | 1.82 | 2.04 | 1.80 | 1.89 |

GUM | $\mathit{\psi}\left(1\right)$ | Irrel. | 6.71 | 1.75 | 1.39 | 1.42 | 5.74 | 6.16 | 5.38 | 5.52 | 2.18 | 3.39 | 2.02 | 2.00 |

$\alpha \phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0$ary | $\mathit{\psi}\left(2\right)$ | Relevant | 54.71 | 43.20 | 6.02 | |||||||||

$\alpha \phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.1$ | $\mathit{\psi}\left(2\right)$ | Relevant | 7.90 | 54.86 | 54.82 | 54.98 | 4.84 | 61.25 | 12.40 | 5.43 | 2.64 | 37.40 | 2.51 | 3.87 |

GUM | $\mathit{\psi}\left(2\right)$ | Relevant | 7.60 | 55.05 | 54.94 | 55.17 | 6.73 | 58.20 | 10.67 | 6.58 | 2.68 | 40.60 | 2.47 | 4.31 |

$\alpha \phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0$ary | $\mathit{\psi}\left(2\right)$ | All | 54.71 | 43.20 | 6.02 | |||||||||

$\alpha \phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.1$ | $\mathit{\psi}\left(2\right)$ | All | 11.05 | 54.86 | 54.82 | 54.98 | 6.76 | 62.80 | 13.90 | 7.23 | 2.65 | 37.15 | 2.59 | 4.27 |

GUM | $\mathit{\psi}\left(2\right)$ | All | 16.45 | 55.05 | 54.94 | 55.17 | 13.99 | 64.88 | 17.65 | 13.72 | 3.02 | 42.09 | 2.71 | 4.41 |

$\alpha \phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0$ary | $\mathit{\psi}\left(2\right)$ | Irrel. | 1.31 | 1.39 | 1.35 | |||||||||

$\alpha \phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.1$ | $\mathit{\psi}\left(2\right)$ | Irrel. | 4.44 | 1.61 | 1.33 | 1.38 | 3.71 | 3.70 | 3.43 | 3.72 | 1.93 | 2.66 | 2.04 | 2.14 |

GUM | $\mathit{\psi}\left(2\right)$ | Irrel. | 10.46 | 1.97 | 1.48 | 1.53 | 8.90 | 9.47 | 8.59 | 8.78 | 2.36 | 4.11 | 2.28 | 2.26 |

$\alpha \phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0$ary | $\mathit{\psi}\left(4\right)$ | Relevant | 54.98 | 39.74 | 5.66 | |||||||||

$\alpha \phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.1$ | $\mathit{\psi}\left(4\right)$ | Relevant | 4.38 | 54.98 | 55.03 | 55.31 | 2.64 | 61.84 | 9.54 | 3.19 | 2.00 | 38.64 | 2.21 | 4.54 |

GUM | $\mathit{\psi}\left(4\right)$ | Relevant | 4.56 | 55.27 | 55.21 | 55.51 | 4.20 | 56.23 | 8.50 | 4.27 | 2.42 | 39.53 | 2.45 | 4.47 |

$\alpha \phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0$ary | $\mathit{\psi}\left(4\right)$ | All | 54.98 | 39.74 | 5.66 | |||||||||

$\alpha \phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.1$ | $\mathit{\psi}\left(4\right)$ | All | 8.45 | 54.98 | 55.03 | 55.31 | 5.27 | 63.59 | 11.82 | 5.71 | 2.31 | 38.55 | 2.55 | 5.00 |

GUM | $\mathit{\psi}\left(4\right)$ | All | 16.47 | 55.27 | 55.21 | 55.51 | 13.89 | 65.37 | 17.89 | 13.79 | 3.00 | 42.11 | 2.85 | 4.59 |

$\alpha \phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0$ary | $\mathit{\psi}\left(4\right)$ | Irrel. | 1.43 | 1.52 | 1.49 | |||||||||

$\alpha \phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.1$ | $\mathit{\psi}\left(4\right)$ | Irrel. | 5.20 | 1.82 | 1.39 | 1.45 | 4.09 | 4.37 | 3.92 | 4.23 | 1.92 | 3.09 | 2.16 | 2.25 |

GUM | $\mathit{\psi}\left(4\right)$ | Irrel. | 13.46 | 2.17 | 1.57 | 1.63 | 11.33 | 12.03 | 11.05 | 11.29 | 2.47 | 4.62 | 2.44 | 2.43 |

**Table 10.**Ratio of MSFE to that of MSFE

_{ary}. Selection at $\alpha =0.1$ for arx, rwx, rdx, and cax.

$\mathit{T}+2|\mathit{T}+1$ | $\mathit{T}+3|\mathit{T}+2$ | $\mathit{T}+4|\mathit{T}+3$ | ||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

arx | rwx | rdx | cax | rwy | cay | arx | rwx | rdx | cax | rwy | cay | arx | rwx | rdx | cax | rwy | cay | |

No break | ||||||||||||||||||

$\mathit{\psi}\left(1\right)$ | 1.10 | 1.15 | 1.31 | 1.15 | 1.21 | 1.29 | 1.11 | 1.16 | 1.31 | 1.16 | 1.21 | 1.29 | 1.10 | 1.14 | 1.30 | 1.15 | 1.21 | 1.28 |

$\mathit{\psi}\left(2\right)$ | 1.00 | 1.04 | 1.24 | 1.05 | 1.14 | 1.19 | 1.02 | 1.07 | 1.29 | 1.08 | 1.15 | 1.22 | 1.01 | 1.06 | 1.25 | 1.06 | 1.15 | 1.21 |

$\mathit{\psi}\left(4\right)$ | 0.96 | 1.00 | 1.23 | 1.01 | 1.12 | 1.16 | 0.97 | 1.02 | 1.26 | 1.03 | 1.12 | 1.17 | 0.96 | 1.00 | 1.23 | 1.00 | 1.12 | 1.17 |

Break in mean and slope (b) of irrelevant regressors | ||||||||||||||||||

$\mathit{\psi}\left(1\right)$ | 2.14 | 2.35 | 3.30 | 2.33 | 1.21 | 1.29 | 1.53 | 1.61 | 1.75 | 1.64 | 1.21 | 1.29 | 1.20 | 1.25 | 1.35 | 1.25 | 1.21 | 1.28 |

$\mathit{\psi}\left(2\right)$ | 2.47 | 2.68 | 3.80 | 2.66 | 1.14 | 1.19 | 1.51 | 1.59 | 1.78 | 1.62 | 1.15 | 1.22 | 1.13 | 1.17 | 1.31 | 1.17 | 1.15 | 1.21 |

$\mathit{\psi}\left(4\right)$ | 2.58 | 2.79 | 3.90 | 2.76 | 1.12 | 1.16 | 1.45 | 1.51 | 1.73 | 1.53 | 1.12 | 1.17 | 1.07 | 1.11 | 1.29 | 1.11 | 1.12 | 1.17 |

Break in mean of all regressors | ||||||||||||||||||

$\mathit{\psi}\left(1\right)$ | 0.62 | 0.50 | 0.34 | 0.50 | 0.63 | 0.57 | 0.48 | 0.47 | 0.75 | 0.48 | 0.28 | 0.26 | 0.72 | 0.69 | 0.82 | 0.69 | 0.67 | 0.67 |

$\mathit{\psi}\left(2\right)$ | 0.57 | 0.42 | 0.25 | 0.42 | 0.69 | 0.62 | 0.50 | 0.58 | 1.19 | 0.60 | 0.37 | 0.34 | 0.79 | 0.84 | 0.92 | 0.85 | 0.85 | 0.85 |

$\mathit{\psi}\left(4\right)$ | 0.54 | 0.37 | 0.22 | 0.37 | 0.72 | 0.65 | 0.51 | 0.69 | 1.61 | 0.72 | 0.43 | 0.40 | 0.81 | 0.96 | 0.98 | 0.96 | 0.94 | 0.94 |

Break in slope (a) of all regressors | ||||||||||||||||||

$\mathit{\psi}\left(1\right)$ | 0.69 | 0.59 | 0.43 | 0.58 | 0.69 | 0.58 | 0.57 | 0.57 | 0.85 | 0.57 | 0.37 | 0.36 | 0.77 | 0.75 | 0.87 | 0.75 | 0.71 | 0.76 |

$\mathit{\psi}\left(2\right)$ | 0.64 | 0.51 | 0.34 | 0.50 | 0.73 | 0.63 | 0.58 | 0.66 | 1.29 | 0.68 | 0.48 | 0.46 | 0.82 | 0.87 | 0.98 | 0.86 | 0.87 | 0.92 |

$\mathit{\psi}\left(4\right)$ | 0.61 | 0.46 | 0.30 | 0.46 | 0.76 | 0.65 | 0.60 | 0.77 | 1.69 | 0.80 | 0.55 | 0.53 | 0.83 | 0.95 | 1.03 | 0.94 | 0.95 | 1.00 |

Break in slope (b) of all regressors | ||||||||||||||||||

$\mathit{\psi}\left(1\right)$ | 0.41 | 0.28 | 0.42 | 0.29 | 0.38 | 0.33 | 0.42 | 0.41 | 0.49 | 0.42 | 0.21 | 0.21 | 0.85 | 0.86 | 0.99 | 0.84 | 1.16 | 1.03 |

$\mathit{\psi}\left(2\right)$ | 0.36 | 0.21 | 0.59 | 0.22 | 0.44 | 0.38 | 0.43 | 0.54 | 0.70 | 0.57 | 0.28 | 0.28 | 0.87 | 1.03 | 1.04 | 0.98 | 1.29 | 1.17 |

$\mathit{\psi}\left(4\right)$ | 0.32 | 0.19 | 0.78 | 0.19 | 0.49 | 0.41 | 0.45 | 0.69 | 0.91 | 0.74 | 0.33 | 0.34 | 0.87 | 1.18 | 1.05 | 1.11 | 1.35 | 1.24 |

Break in mean and slope (a) of all regressors | ||||||||||||||||||

$\mathit{\psi}\left(1\right)$ | 0.83 | 0.78 | 0.75 | 0.78 | 0.86 | 0.87 | 0.88 | 0.91 | 1.11 | 0.92 | 0.79 | 0.82 | 0.99 | 1.01 | 1.14 | 1.01 | 1.00 | 1.05 |

$\mathit{\psi}\left(2\right)$ | 0.76 | 0.69 | 0.67 | 0.69 | 0.86 | 0.87 | 0.87 | 0.94 | 1.31 | 0.95 | 0.86 | 0.88 | 0.97 | 1.01 | 1.17 | 1.01 | 1.06 | 1.10 |

$\mathit{\psi}\left(4\right)$ | 0.73 | 0.65 | 0.63 | 0.64 | 0.87 | 0.87 | 0.85 | 0.95 | 1.42 | 0.96 | 0.88 | 0.91 | 0.93 | 1.00 | 1.18 | 1.00 | 1.08 | 1.10 |

Break in mean and slope (b) of all regressors | ||||||||||||||||||

$\mathit{\psi}\left(1\right)$ | 0.37 | 0.23 | 0.39 | 0.25 | 0.35 | 0.32 | 0.43 | 0.53 | 0.67 | 0.60 | 0.21 | 0.22 | 0.86 | 1.09 | 1.07 | 1.03 | 1.64 | 1.44 |

$\mathit{\psi}\left(2\right)$ | 0.32 | 0.17 | 0.55 | 0.18 | 0.42 | 0.37 | 0.43 | 0.71 | 0.94 | 0.82 | 0.26 | 0.27 | 0.83 | 1.25 | 1.04 | 1.17 | 1.66 | 1.51 |

$\mathit{\psi}\left(4\right)$ | 0.30 | 0.14 | 0.71 | 0.16 | 0.46 | 0.40 | 0.45 | 0.88 | 1.19 | 1.04 | 0.30 | 0.31 | 0.82 | 1.41 | 1.02 | 1.30 | 1.67 | 1.55 |

Average over all breaks in all regressors | ||||||||||||||||||

$\mathit{\psi}\left(1\right)$ | 0.58 | 0.48 | 0.47 | 0.48 | 0.58 | 0.54 | 0.56 | 0.58 | 0.77 | 0.60 | 0.37 | 0.37 | 0.84 | 0.88 | 0.98 | 0.87 | 1.04 | 0.99 |

$\mathit{\psi}\left(2\right)$ | 0.53 | 0.40 | 0.48 | 0.40 | 0.63 | 0.58 | 0.56 | 0.69 | 1.08 | 0.72 | 0.45 | 0.45 | 0.86 | 1.00 | 1.03 | 0.97 | 1.15 | 1.11 |

$\mathit{\psi}\left(4\right)$ | 0.50 | 0.36 | 0.53 | 0.36 | 0.66 | 0.60 | 0.57 | 0.80 | 1.37 | 0.85 | 0.50 | 0.50 | 0.85 | 1.10 | 1.05 | 1.06 | 1.20 | 1.17 |

$\mathit{T}+2|\mathit{T}+1$ | $\mathit{T}+3|\mathit{T}+2$ | $\mathit{T}+4|\mathit{T}+3$ | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

inf | arx | rwx | rdx | cax | inf | arx | rwx | rdx | cax | inf | arx | rwx | rdx | cax | |

No break in y: no break and break in irrelevant variables | |||||||||||||||

$\alpha =0.01$ | 1.13 | 1.15 | 1.22 | 1.55 | 1.22 | 0.99 | 1.07 | 1.13 | 1.26 | 1.13 | 0.93 | 1.01 | 1.04 | 1.16 | 1.04 |

$\alpha =0.05$ | 1.52 | 1.47 | 1.59 | 2.14 | 1.57 | 1.14 | 1.21 | 1.27 | 1.45 | 1.28 | 0.99 | 1.05 | 1.10 | 1.25 | 1.10 |

$\alpha =0.1$ | 1.78 | 1.71 | 1.84 | 2.46 | 1.81 | 1.21 | 1.26 | 1.33 | 1.52 | 1.33 | 1.03 | 1.08 | 1.12 | 1.29 | 1.12 |

GUM | 3.69 | 3.55 | 3.64 | 4.17 | 3.61 | 1.46 | 1.41 | 1.42 | 1.60 | 1.41 | 1.21 | 1.17 | 1.19 | 1.41 | 1.19 |

DGP | 0.82 | 0.92 | 0.96 | 1.12 | 0.96 | 0.83 | 0.93 | 0.97 | 1.14 | 0.97 | 0.82 | 0.93 | 0.96 | 1.13 | 0.96 |

Break in y: break in all variables | |||||||||||||||

$\alpha =0.01$ | 0.40 | 0.63 | 0.52 | 0.52 | 0.52 | 0.59 | 0.62 | 0.68 | 0.94 | 0.70 | 0.82 | 0.87 | 0.99 | 1.00 | 0.97 |

$\alpha =0.05$ | 0.31 | 0.56 | 0.43 | 0.48 | 0.44 | 0.54 | 0.56 | 0.67 | 1.02 | 0.70 | 0.82 | 0.85 | 0.99 | 1.01 | 0.96 |

$\alpha =0.1$ | 0.30 | 0.54 | 0.41 | 0.49 | 0.42 | 0.55 | 0.56 | 0.69 | 1.07 | 0.72 | 0.83 | 0.85 | 0.99 | 1.02 | 0.97 |

GUM | 0.38 | 0.54 | 0.44 | 0.64 | 0.43 | 0.67 | 0.65 | 0.79 | 1.26 | 0.83 | 0.96 | 0.91 | 1.00 | 1.09 | 0.98 |

DGP | 0.17 | 0.44 | 0.29 | 0.41 | 0.30 | 0.36 | 0.40 | 0.61 | 1.11 | 0.66 | 0.64 | 0.72 | 1.02 | 0.86 | 0.97 |

**Table 12.**Ratio of MSFE to that of MSFE

_{ary}. Selection at $\alpha =0.1$. Average over noncentralities and horizons $T+2,$ … $,T+4$. Lowest two in bold (excluding inf).

inf | avg | arx | rwx | rdx | cax | rwy | cay | apool | cpool | ary | |
---|---|---|---|---|---|---|---|---|---|---|---|

No break | 0.99 | 1.22 | 1.03 | 1.07 | 1.27 | 1.07 | 1.16 | 1.22 | 0.96 | 1.00 | 1.00 |

Break irrelevant | 1.69 | 1.99 | 1.68 | 1.78 | 2.25 | 1.77 | 1.16 | 1.22 | 1.13 | 1.20 | 1.00 |

All breaks | 0.56 | 2.93 | 0.65 | 0.70 | 0.86 | 0.70 | 0.73 | 0.70 | 0.73 | 0.58 | 1.00 |

Sum | 3.24 | 6.14 | 3.36 | 3.55 | 4.38 | 3.54 | 3.05 | 3.14 | 2.82 | 2.78 | 3.00 |

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**MDPI and ACS Style**

Castle, J.L.; Doornik, J.A.; Hendry, D.F.
Selecting a Model for Forecasting. *Econometrics* **2021**, *9*, 26.
https://doi.org/10.3390/econometrics9030026

**AMA Style**

Castle JL, Doornik JA, Hendry DF.
Selecting a Model for Forecasting. *Econometrics*. 2021; 9(3):26.
https://doi.org/10.3390/econometrics9030026

**Chicago/Turabian Style**

Castle, Jennifer L., Jurgen A. Doornik, and David F. Hendry.
2021. "Selecting a Model for Forecasting" *Econometrics* 9, no. 3: 26.
https://doi.org/10.3390/econometrics9030026