Forecasting

library(bvhar)

Simulation

Given VAR coefficient and VHAR coefficient each,

We use coefficient matrix estimated by VAR(5) in introduction vignette.

Consider

coef(ex_fit)
#>              GVZCLS   OVXCLS    EVZCLS VXFXICLS
#> GVZCLS_1    0.93302 -0.02332 -0.007712 -0.03853
#> OVXCLS_1    0.05429  1.00399  0.009806  0.01062
#> EVZCLS_1    0.06794 -0.13900  0.983825  0.07783
#> VXFXICLS_1 -0.03399  0.03404  0.020719  0.93350
#> GVZCLS_2   -0.07831  0.08753  0.019302  0.08939
#> OVXCLS_2   -0.04770  0.01480  0.003888  0.04392
#> EVZCLS_2    0.08082  0.26704 -0.110017 -0.07163
#> VXFXICLS_2  0.05465 -0.12154 -0.040349  0.04012
#> GVZCLS_3    0.04332 -0.02459 -0.011041 -0.02556
#> OVXCLS_3   -0.00594 -0.09550  0.006638 -0.04981
#> EVZCLS_3   -0.02952 -0.04926  0.091056  0.01204
#> VXFXICLS_3 -0.05876 -0.05995  0.003803 -0.02027
#> GVZCLS_4   -0.00845 -0.04490  0.005415 -0.00817
#> OVXCLS_4    0.01070 -0.00383 -0.022806 -0.05557
#> EVZCLS_4   -0.01971 -0.02008 -0.016535  0.08229
#> VXFXICLS_4  0.06139  0.14403  0.019780 -0.10271
#> GVZCLS_5    0.07301  0.01093 -0.010994 -0.01526
#> OVXCLS_5   -0.01658  0.07401  0.007035  0.04297
#> EVZCLS_5   -0.08794 -0.06189  0.021082 -0.02465
#> VXFXICLS_5 -0.01739  0.00169  0.000335  0.09384
#> const       0.57370  0.15256  0.132842  0.87785
ex_fit$covmat
#>          GVZCLS OVXCLS EVZCLS VXFXICLS
#> GVZCLS    1.157  0.403  0.127    0.332
#> OVXCLS    0.403  1.740  0.115    0.438
#> EVZCLS    0.127  0.115  0.144    0.127
#> VXFXICLS  0.332  0.438  0.127    1.028

Then

m <- ncol(ex_fit$coefficients)
# generate VAR(5)-----------------
y <- sim_var(
  num_sim = 1500, 
  num_burn = 100, 
  var_coef = coef(ex_fit), 
  var_lag = 5L, 
  sig_error = ex_fit$covmat, 
  init = matrix(0L, nrow = 5L, ncol = m)
)
# colname: y1, y2, ...------------
colnames(y) <- paste0("y", 1:m)
head(y)
#>        y1   y2   y3   y4
#> [1,] 18.7 26.5 7.55 26.2
#> [2,] 18.2 25.8 7.39 25.6
#> [3,] 19.7 25.3 7.40 26.1
#> [4,] 20.6 24.5 7.34 26.4
#> [5,] 21.6 24.6 7.06 27.8
#> [6,] 22.5 23.7 7.02 25.8
h <- 20
y_eval <- divide_ts(y, h)
y_train <- y_eval$train # train
y_test <- y_eval$test # test

Fitting Models

VAR(5) and VHAR

# VAR(5)
model_var <- var_lm(y_train, 5)
# VHAR
model_vhar <- vhar_lm(y_train)

BVAR(5)

Minnesota prior

# hyper parameters---------------------------
y_sig <- apply(y_train, 2, sd) # sigma vector
y_lam <- .2 # lambda
y_delta <- rep(.2, m) # delta vector (0 vector since RV stationary)
eps <- 1e-04 # very small number
spec_bvar <- set_bvar(y_sig, y_lam, y_delta, eps)
# fit---------------------------------------
model_bvar <- bvar_minnesota(y_train, p = 5, bayes_spec = spec_bvar)

BVHAR

BVHAR-S

spec_bvhar_v1 <- set_bvhar(y_sig, y_lam, y_delta, eps)
# fit---------------------------------------
model_bvhar_v1 <- bvhar_minnesota(y_train, bayes_spec = spec_bvhar_v1)

BVHAR-L

# weights----------------------------------
y_day <- rep(.1, m)
y_week <- rep(.01, m)
y_month <- rep(.01, m)
# spec-------------------------------------
spec_bvhar_v2 <- set_weight_bvhar(
  y_sig,
  y_lam,
  eps,
  y_day,
  y_week,
  y_month
)
# fit--------------------------------------
model_bvhar_v2 <- bvhar_minnesota(y_train, bayes_spec = spec_bvhar_v2)

Splitting

You can forecast using predict() method with above objects. You should set the step of the forecasting using n_ahead argument.

In addition, the result of this forecast will return another class called predbvhar to use some methods,

VAR

(pred_var <- predict(model_var, n_ahead = h))
#>         y1   y2   y3   y4
#>  [1,] 17.0 37.3 9.56 22.2
#>  [2,] 16.8 37.4 9.56 22.4
#>  [3,] 16.7 37.3 9.58 22.5
#>  [4,] 16.7 37.2 9.57 22.6
#>  [5,] 16.7 37.1 9.58 22.7
#>  [6,] 16.6 37.0 9.58 22.7
#>  [7,] 16.6 36.9 9.58 22.8
#>  [8,] 16.5 36.8 9.59 22.9
#>  [9,] 16.5 36.8 9.59 22.9
#> [10,] 16.4 36.7 9.59 23.0
#> [11,] 16.4 36.6 9.60 23.1
#> [12,] 16.3 36.5 9.60 23.1
#> [13,] 16.3 36.4 9.60 23.2
#> [14,] 16.3 36.3 9.60 23.3
#> [15,] 16.2 36.3 9.61 23.3
#> [16,] 16.2 36.2 9.61 23.4
#> [17,] 16.2 36.1 9.61 23.4
#> [18,] 16.1 36.0 9.61 23.5
#> [19,] 16.1 35.9 9.61 23.5
#> [20,] 16.1 35.9 9.61 23.6
class(pred_var)
#> [1] "predbvhar"
names(pred_var)
#> [1] "process"     "forecast"    "se"          "lower"       "upper"      
#> [6] "lower_joint" "upper_joint" "y"

The package provides the evaluation function

(mse_var <- mse(pred_var, y_test))
#>     y1     y2     y3     y4 
#>  2.416 22.739  0.372  3.115

VHAR

(pred_vhar <- predict(model_vhar, n_ahead = h))
#>         y1   y2   y3   y4
#>  [1,] 17.0 37.5 9.57 22.4
#>  [2,] 16.9 37.4 9.56 22.5
#>  [3,] 16.8 37.3 9.55 22.5
#>  [4,] 16.7 37.2 9.54 22.5
#>  [5,] 16.6 37.2 9.53 22.6
#>  [6,] 16.5 37.1 9.52 22.6
#>  [7,] 16.4 37.0 9.51 22.6
#>  [8,] 16.3 36.9 9.49 22.6
#>  [9,] 16.2 36.9 9.48 22.6
#> [10,] 16.2 36.8 9.46 22.6
#> [11,] 16.1 36.7 9.45 22.7
#> [12,] 16.0 36.7 9.43 22.7
#> [13,] 15.9 36.6 9.42 22.7
#> [14,] 15.9 36.6 9.41 22.7
#> [15,] 15.8 36.5 9.40 22.8
#> [16,] 15.8 36.5 9.40 22.8
#> [17,] 15.7 36.4 9.39 22.9
#> [18,] 15.7 36.4 9.39 22.9
#> [19,] 15.7 36.3 9.39 23.0
#> [20,] 15.6 36.3 9.39 23.0

MSE:

(mse_vhar <- mse(pred_vhar, y_test))
#>    y1    y2    y3    y4 
#>  3.29 24.46  0.27  3.05

BVAR

(pred_bvar <- predict(model_bvar, n_ahead = h))
#>         y1   y2   y3   y4
#>  [1,] 17.0 37.4 9.52 22.4
#>  [2,] 17.0 37.3 9.51 22.6
#>  [3,] 16.9 37.1 9.51 22.7
#>  [4,] 16.8 37.0 9.51 22.8
#>  [5,] 16.8 36.9 9.51 22.9
#>  [6,] 16.7 36.8 9.52 22.9
#>  [7,] 16.7 36.7 9.52 23.0
#>  [8,] 16.6 36.6 9.52 23.1
#>  [9,] 16.6 36.5 9.52 23.2
#> [10,] 16.5 36.3 9.53 23.3
#> [11,] 16.5 36.2 9.53 23.3
#> [12,] 16.4 36.1 9.53 23.4
#> [13,] 16.4 36.0 9.53 23.5
#> [14,] 16.4 35.9 9.53 23.5
#> [15,] 16.3 35.8 9.53 23.6
#> [16,] 16.3 35.7 9.54 23.6
#> [17,] 16.3 35.6 9.54 23.7
#> [18,] 16.2 35.5 9.54 23.8
#> [19,] 16.2 35.4 9.54 23.8
#> [20,] 16.2 35.3 9.54 23.9

MSE:

(mse_bvar <- mse(pred_bvar, y_test))
#>     y1     y2     y3     y4 
#>  2.202 19.792  0.319  3.414

BVHAR

VAR-type Minnesota

(pred_bvhar_v1 <- predict(model_bvhar_v1, n_ahead = h))
#>         y1   y2   y3   y4
#>  [1,] 16.9 37.4 9.53 22.4
#>  [2,] 16.9 37.2 9.50 22.5
#>  [3,] 16.8 37.1 9.48 22.5
#>  [4,] 16.8 36.9 9.47 22.6
#>  [5,] 16.7 36.8 9.46 22.7
#>  [6,] 16.6 36.7 9.45 22.7
#>  [7,] 16.5 36.6 9.44 22.8
#>  [8,] 16.5 36.5 9.43 22.8
#>  [9,] 16.4 36.4 9.43 22.8
#> [10,] 16.3 36.3 9.42 22.9
#> [11,] 16.3 36.2 9.41 22.9
#> [12,] 16.2 36.1 9.41 23.0
#> [13,] 16.2 36.0 9.40 23.0
#> [14,] 16.1 36.0 9.40 23.1
#> [15,] 16.1 35.9 9.40 23.1
#> [16,] 16.1 35.8 9.40 23.2
#> [17,] 16.0 35.8 9.40 23.2
#> [18,] 16.0 35.7 9.40 23.3
#> [19,] 16.0 35.6 9.40 23.3
#> [20,] 16.0 35.5 9.40 23.4

MSE:

(mse_bvhar_v1 <- mse(pred_bvhar_v1, y_test))
#>     y1     y2     y3     y4 
#>  2.655 19.914  0.256  3.103

VHAR-type Minnesota

(pred_bvhar_v2 <- predict(model_bvhar_v2, n_ahead = h))
#>         y1   y2   y3   y4
#>  [1,] 16.9 37.4 9.53 22.4
#>  [2,] 16.9 37.2 9.50 22.5
#>  [3,] 16.8 37.0 9.47 22.5
#>  [4,] 16.8 36.9 9.46 22.6
#>  [5,] 16.7 36.8 9.45 22.6
#>  [6,] 16.6 36.7 9.44 22.7
#>  [7,] 16.5 36.6 9.43 22.7
#>  [8,] 16.5 36.5 9.43 22.8
#>  [9,] 16.4 36.4 9.42 22.8
#> [10,] 16.4 36.3 9.41 22.9
#> [11,] 16.3 36.2 9.40 22.9
#> [12,] 16.2 36.1 9.40 22.9
#> [13,] 16.2 36.0 9.39 23.0
#> [14,] 16.2 35.9 9.39 23.0
#> [15,] 16.1 35.9 9.39 23.1
#> [16,] 16.1 35.8 9.39 23.1
#> [17,] 16.0 35.7 9.39 23.2
#> [18,] 16.0 35.7 9.39 23.2
#> [19,] 16.0 35.6 9.39 23.3
#> [20,] 16.0 35.5 9.39 23.3

MSE:

(mse_bvhar_v2 <- mse(pred_bvhar_v2, y_test))
#>     y1     y2     y3     y4 
#>  2.630 19.668  0.252  3.095

Compare

Region

autoplot(predbvhar) and autolayer(predbvhar) draws the results of the forecasting.

autoplot(pred_var, x_cut = 1470, ci_alpha = .7, type = "wrap") +
  autolayer(pred_vhar, ci_alpha = .5) +
  autolayer(pred_bvar, ci_alpha = .4) +
  autolayer(pred_bvhar_v1, ci_alpha = .2) +
  autolayer(pred_bvhar_v2, ci_alpha = .1) +
  geom_eval(y_test, colour = "#000000", alpha = .5)

Error

Mean of MSE

list(
  VAR = mse_var,
  VHAR = mse_vhar,
  BVAR = mse_bvar,
  BVHAR1 = mse_bvhar_v1,
  BVHAR2 = mse_bvhar_v2
) %>% 
  lapply(mean) %>% 
  unlist() %>% 
  sort()
#> BVHAR2   BVAR BVHAR1    VAR   VHAR 
#>   6.41   6.43   6.48   7.16   7.77

For each variable, we can see the error with plot.

list(
  pred_var,
  pred_vhar,
  pred_bvar,
  pred_bvhar_v1,
  pred_bvhar_v2
) %>% 
  gg_loss(y = y_test, "mse")

Relative MAPE (MAPE), benchmark model: VAR

list(
  VAR = pred_var,
  VHAR = pred_vhar,
  BVAR = pred_bvar,
  BVHAR1 = pred_bvhar_v1,
  BVHAR2 = pred_bvhar_v2
) %>% 
  lapply(rmape, pred_bench = pred_var, y = y_test) %>% 
  unlist()
#>    VAR   VHAR   BVAR BVHAR1 BVHAR2 
#>  1.000  1.020  0.965  0.954  0.948

Out-of-Sample Forecasting

In time series research, out-of-sample forecasting plays a key role. So, we provide out-of-sample forecasting function based on

Rolling windows

forecast_roll(object, n_ahead, y_test) conducts h >= 1 step rolling windows forecasting.

It fixes window size and moves the window. The window is the training set. In this package, we set window size = original input data.

Iterating the step

  1. The model is fitted in the training set.
  2. With the fitted model, researcher should forecast the next h >= 1 step ahead. The longest forecast horizon is num_test - h + 1.
  3. After this window, move the window and do the same process.
  4. Get forecasted values until possible (longest forecast horizon).

5-step out-of-sample:

(var_roll <- forecast_roll(model_var, 5, y_test))
#>         y1   y2   y3   y4
#>  [1,] 16.7 37.1 9.58 22.7
#>  [2,] 17.6 34.9 9.48 23.4
#>  [3,] 16.7 35.0 9.73 22.5
#>  [4,] 16.6 32.5 8.98 21.7
#>  [5,] 16.0 31.6 8.83 22.3
#>  [6,] 16.5 32.9 8.64 22.6
#>  [7,] 17.1 32.9 9.12 22.8
#>  [8,] 17.5 32.2 9.27 22.5
#>  [9,] 17.5 30.7 9.57 22.1
#> [10,] 18.5 32.8 9.93 22.2
#> [11,] 18.2 31.6 9.67 21.5
#> [12,] 18.2 30.5 9.47 22.6
#> [13,] 18.1 30.9 9.19 21.5
#> [14,] 17.3 30.7 8.83 21.0
#> [15,] 19.0 31.3 9.18 23.2
#> [16,] 17.6 31.1 8.71 22.9

Denote that the nrow is longest forecast horizon.

class(var_roll)
#> [1] "predbvhar_roll" "bvharcv"
names(var_roll)
#> [1] "process"  "forecast" "eval_id"  "y"

To apply the same evaluation methods, a class named bvharcv has been defined. You can use the functions above.

vhar_roll <- forecast_roll(model_vhar, 5, y_test)
bvar_roll <- forecast_roll(model_bvar, 5, y_test)
bvhar_roll_v1 <- forecast_roll(model_bvhar_v1, 5, y_test)
bvhar_roll_v2 <- forecast_roll(model_bvhar_v2, 5, y_test)

Relative MAPE, benchmark model: VAR

list(
  VAR = var_roll,
  VHAR = vhar_roll,
  BVAR = bvar_roll,
  BVHAR1 = bvhar_roll_v1,
  BVHAR2 = bvhar_roll_v2
) %>% 
  lapply(rmape, pred_bench = var_roll, y = y_test) %>% 
  unlist()
#>    VAR   VHAR   BVAR BVHAR1 BVHAR2 
#>  1.000  0.989  0.982  0.973  0.973

Expanding Windows

forecast_expand(object, n_ahead, y_test) conducts h >= 1 step expanding window forecasting.

Different with rolling windows, expanding windows method fixes the starting point. The other is same.

(var_expand <- forecast_expand(model_var, 5, y_test))
#>         y1   y2   y3   y4
#>  [1,] 16.7 37.1 9.58 22.7
#>  [2,] 17.6 34.9 9.48 23.4
#>  [3,] 16.7 35.0 9.73 22.5
#>  [4,] 16.6 32.4 8.97 21.7
#>  [5,] 16.0 31.6 8.82 22.3
#>  [6,] 16.5 32.9 8.63 22.6
#>  [7,] 17.1 32.9 9.12 22.8
#>  [8,] 17.5 32.2 9.27 22.5
#>  [9,] 17.5 30.7 9.58 22.1
#> [10,] 18.5 32.8 9.95 22.2
#> [11,] 18.2 31.6 9.68 21.5
#> [12,] 18.2 30.5 9.48 22.6
#> [13,] 18.1 30.9 9.19 21.5
#> [14,] 17.3 30.7 8.84 21.0
#> [15,] 19.0 31.3 9.17 23.2
#> [16,] 17.6 31.1 8.70 22.9

The class is bvharcv.

class(var_expand)
#> [1] "predbvhar_expand" "bvharcv"
names(var_expand)
#> [1] "process"  "forecast" "eval_id"  "y"
vhar_expand <- forecast_expand(model_vhar, 5, y_test)
bvar_expand <- forecast_expand(model_bvar, 5, y_test)
bvhar_expand_v1 <- forecast_expand(model_bvhar_v1, 5, y_test)
bvhar_expand_v2 <- forecast_expand(model_bvhar_v2, 5, y_test)

Relative MAPE, benchmark model: VAR

list(
  VAR = var_expand,
  VHAR = vhar_expand,
  BVAR = bvar_expand,
  BVHAR1 = bvhar_expand_v1,
  BVHAR2 = bvhar_expand_v2
) %>% 
  lapply(rmape, pred_bench = var_expand, y = y_test) %>% 
  unlist()
#>    VAR   VHAR   BVAR BVHAR1 BVHAR2 
#>  1.000  0.985  0.982  0.969  0.969