Forecasting with Vector Autoregression
- 2. What is Vector Autoregression • Vector autoregression (VAR) is a time series method used when two time series interact and help predict each other. • A sample of a 2-dimensional VAR(1) is shown below • The vector can be resolved into a system of equations to predict each other
- 3. Binds and Leads Represent Two Series that Interact • Original Series from July to December • January data is used for testing the data • Total Binds and Outbound Leads used • Removed Saturday, Sunday and holidays
- 4. Cross Correlogram Showing Relationship Between Leads and Binds • At positive lag values, Binds lags behind Leads • Binds = Leads (t+1) • At negative lag values, Binds proceeds Leads • Binds = Leads (t-1) • Values above the dotted blue lines indicate positive correlation and significance • Data suggest that a 5 period model is best • Captures weekly periodicity of production • Strong correlations for up to 10 days Binds are Leading Binds are Lagging
- 5. Model Output (2 Equations – Binds, Leads for 5 Lags + Constant) Estimation results for equation Leads: ====================================== Leads = Binds.l1 + Leads.l1 + Binds.l2 + Leads.l2 + Binds.l3 + Leads.l3 + Binds.l4 + Leads.l4 + Binds.l5 + Leads.l5 + const Estimate Std. Error t value Pr(>|t|) Binds.l1 2.21335 1.18112 1.874 0.0633 . Leads.l1 0.58684 0.09207 6.374 3.43e-09 *** Binds.l2 0.44731 1.17360 0.381 0.7038 Leads.l2 0.16763 0.10535 1.591 0.1141 Binds.l3 0.06551 1.14555 0.057 0.9545 Leads.l3 0.02270 0.10539 0.215 0.8298 Binds.l4 2.12845 1.15697 1.840 0.0682 . Leads.l4 -0.03806 0.10252 -0.371 0.7111 Binds.l5 1.06536 1.17110 0.910 0.3648 Leads.l5 0.07396 0.08357 0.885 0.3779 const -263.92014 502.30505 -0.525 0.6002 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 719 on 122 degrees of freedom Multiple R-Squared: 0.8871 Adjusted R-squared: 0.8779 F-statistic: 95.87 on 10 and 122 DF, p-value: < 2.2e-16 Estimation results for equation Binds: ====================================== Binds = Binds.l1 + Leads.l1 + Binds.l2 + Leads.l2 + Binds.l3 + Leads.l3 + Binds.l4 + Leads.l4 + Binds.l5 + Leads.l5 + const Estimate Std. Error t value Pr(>|t|) Binds.l1 0.101506 0.082297 1.233 0.21979 Leads.l1 0.013114 0.006415 2.044 0.04309 * Binds.l2 -0.029857 0.081773 -0.365 0.71565 Leads.l2 0.007010 0.007340 0.955 0.34145 Binds.l3 -0.138045 0.079818 -1.729 0.08625 . Leads.l3 -0.007207 0.007343 -0.981 0.32831 Binds.l4 0.101189 0.080614 1.255 0.21179 Leads.l4 -0.001406 0.007143 -0.197 0.84427 Binds.l5 0.468908 0.081598 5.747 6.85e-08 *** Leads.l5 -0.006999 0.005823 -1.202 0.23174 const 106.246711 34.998936 3.036 0.00293 ** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 50.1 on 122 degrees of freedom Multiple R-Squared: 0.4322 Adjusted R-squared: 0.3857 F-statistic: 9.288 on 10 and 122 DF, p-value: 2.6e-11 Estimation results for equation Leads: ====================================== Leads = Binds.l1 + Leads.l1 + Binds.l2 + Leads.l2 + Binds.l3 + Leads.l3 + Binds.l4
- 6. Model Diagnostics For Binds • Model is a good fit for testing data • Residuals show random behavior • Very little structure left in autocorrelations and partial autocorrelations
- 7. Model Diagnostics for Leads • Model is a good fit for testing data • Residuals show random behavior • Very little structure left in autocorrelations and partial autocorrelations
- 8. Forecasting with the Model
- 9. Comparison of Forecasts to Actual January Data Binds Forecast Actual Delta (Act - Fcast) % 1 279 307 28 9% 2 297 324 27 8% 3 259 223 -36 -16% 4 244 265 21 8% 5 247 286 39 14% 6 278 297 19 6% 7 289 327 38 12% 8 271 215 -56 -26% 9 259 240 -19 -8% 10 260 295 35 12% 11 278 350 72 21% Total 2961 3129 168 5% Leads Forecast Actual Delta (Act - Fcast) % 1 7,055 7,297 242 3.32% 2 7,038 7,290 252 3.46% 3 6,992 7,043 51 0.72% 4 6,907 7,120 213 2.99% 5 6,880 7,083 203 2.87% 6 6,952 7,041 89 1.26% 7 6,992 6,967 -25 -0.36% 8 6,994 7,080 86 1.21% 9 6,956 7,096 140 1.97% 10 6,964 7,231 267 3.69% 11 7,020 6,976 -44 -0.63% Total 76,750 78,224 1474 1.88%
- 10. Conclusions • Leads and Binds are two series that can be used to forecast each other • Using the interaction, 5 previous periods are required to forecast 1 period forward • There exists some daily volatility in binds, but aggregate binds over a two week period is smooth and can be forecasted with a 5% error • Model forecast error for binds up to 10 days out is 5% • Model forecast error for associated leads is less than 2% • Model is capable of “learning” as our leads and binds pattern changes • Opportunities for improvements or other model choices • Other SVAR models • Simpler ARIMA models • ARCH/GARCH • Addition of other exogenous variables