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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

These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.