Rolling Origin via ARDL model in R

TalhaAsif · August 20, 2022, 1:11am

I have this dataset where I want to use rolling origin as a validation technique to measure and observe the best model based on the error measure. I have tried to implement the Rolling origin method in my ARDL model but have had no success. Would like to know if someone can point me in the right direction.

Data:

structure(list(Industrialproduction = c(1.65801981343852, 1.79541527049647, 
-0.0326429293424051, 0.104752527715549, -0.992082392187777, -2.26823002723453, 
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0.104752527715549, 0.228175565411677, 0.305023871901719, 0.218860619170459, 
0.216531882610155, 0.139683576120113, 0.25146293101472, 0.249134194454415, 
0.626389517223712, 1.13405408737005, 0.58214352257793, -0.0629165046263609, 
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1.73021064680795, 0.561184893535192, 0.207216936368938, -0.489075295162048
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), Investmentgrowth = c(1.47348593810751, 2.17792802452104, 2.57620375293532, 
3.11977876989162, 2.03003410582649, 1.238671909303, 0.670447905897604, 
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-1.12853656861309, -1.1210277646243, -1.11622213007147, -1.09820100049835, 
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1.52084996657255, 1.52084996657255, 2.1284604529957, 2.10623080105339, 
1.8221852484572, 1.74561644732258, 1.8221852484572, 1.79254571253412, 
2.04695172920723, 2.32358739782265, 2.3705166630342, 1.87899435897644, 
1.81477536447643, 1.26150402724559, 1.27879375653405, 1.22939452999558, 
1.13306603824557, 1.19728503274558, 1.07872688905325, 1.0515573144571, 
1.28126371786097, 1.19234511009173, 1.10342650232249, 1.21704472336097, 
1.14294588355326, 1.05649723711094, 1.08119685038018, 0.811971065745526, 
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2.09326712428386, 1.92386866579365, 1.57025981463491, 1.25236650053715, 
1.3619748134568, 1.79713275011286, 1.62283175002842, 1.46140252936374, 
1.11755703763521, 0.767494705927617, 1.00504892351727, 0.836192873884299, 
0.743461821176297, 0.639548801257612, 0.614285065637952, 0.59615609267884, 
1.0233656533885, 0.831874464905546, 1.44483819733896, 1.35112663631211, 
1.16530987895099, 1.12197975600959, 1.27587783831036, 0.9050787987531
), ConsumerConfidenceIndex = c(0.846829650502804, 1.60472118016078, 
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1.18224076468833, 1.53172580694677, 1.23795881575313, 0.203420073079754, 
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-0.38459179268613, -0.144363494292252, -0.345373846847121, -0.204466519215588
), RealPersonalDisposableIncome = c(-1.61847984374121, -1.5861635599299, 
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1.64487557992411, -2.69709543568468, 3.49680170575694, 3.25504738360115, 
2.39425379090184, 2.98519095869059, 4.36691137516082, 3.57868020304568, 
1.66275772744776, 3.79450451070863, 4.52162951167727, 2.28203256419209, 
4.17054552224914, 3.2439678284182, 4.76643873164257, 0.955633279171614, 
2.91614381581101, 0.848198902642676, 5.02010671012167), OUTERSEAST = c(6.7110371602884, 
7.53638253638255, 9.47317544707589, 8.56512141280351, 3.82269215128102, 
2.11515863689776, 1.64940544687381, -1.73584905660378, 1.34408602150539, 
1.78097764304659, 0.446760982874161, -1.26019273535953, 0.150150150150159, 
3.11094452773611, 1.4176663031625, 2.54480286738352, 5.56448794127927, 
4.89371564797033, 3.88257575757575, 1.85961713764815, 5.54859495256845, 
4.29879599796508, 2.00525702517411, 3.63679834232127, 3.44509381728699, 
3.46664684309643, 1.93988743863012, 2.50440502760482, 2.96578121060713, 
4.47634947134114), OUTERMET = c(4.54545454545458, 6.58505698607005, 
7.36633663366336, 7.08225746956843, 4.3747847054771, 1.68316831683168, 
1.00616682895164, -1.28534704370181, 2.01822916666665, 0.797702616464613, 
0.949667616334271, -0.940733772342415, 1.10794555238999, 2.19160926737633, 
2.84926470588237, 2.62138814417631, 5.02467343976781, 5.65213786241397, 
3.22555328833776, 3.73552294786995, 5.05948745510956, 4.28797321179426, 
2.86300392436674, 2.60339894216597, 4.28031183318191, 3.43199821714381, 
3.34554286721641, 3.04770569170409, 1.65167650683293, 4.62120252591965
), LONDON = c(8.11719500480309, 10.3065304309196, 6.32299637535239, 
7.65151515151515, 1.30190007037299, 2.1535255296978, -0.204012240734436, 
-0.306643952299836, 0.786056049213951, 1.18684299762631, 1.00536193029493, 
-2.85335102853352, 2.76639344262296, 2.06048521103356, 1.23738196027352, 
2.70183338694115, 3.30410272471031, 5.76322570865546, 4.73255747291176, 
1.98428989791171, 6.03563952552197, 4.88977753030802, 2.12581135535556, 
4.43247330120026, 5.42986425339366, 3.96781115879828, 3.43247538648888, 
4.0668901660281, 4.09587727708534, 4.81707991010573), SOUTHWEST = c(6.17577197149644, 
7.71812080536912, 7.63239875389407, 9.45489628557649, 2.46804759806079, 
2.19354838709679, 1.72558922558922, 0.248241621845247, 1.48576145274456, 
2.03334688897925, -0.677560781187733, -2.3274478330658, 1.80772391125718, 
2.42130750605327, 1.85185185185186, 0.928433268858785, 5.95247221157533, 
4.38447346525341, 3.30272049904696, 2.25107353730542, 3.86823714688802, 
2.04371722787289, 3.04596811639065, 4.19057346270538, 2.45646407565451, 
2.17525889239081, 2.83400809716597, 1.58015962290428, 2.77894958869438, 
4.08650146221331), WALES = c(6.09418282548476, 8.35509138381203, 
7.40963855421687, 7.01065619742007, 1.15303983228513, 3.47150259067357, 
-0.150225338007013, 0.852557673019058, 0.944803580308295, -1.13300492610835, 
0.946686596910786, -2.17176702862782, 3.98587285570131, 0.485201358563789, 
3.62143891839691, 1.63094128611373, 1.61852361302152, 4.32251951450617, 
1.28887158859911, 0.68747598104105, 3.71925360474978, 4.66941979801284, 
1.44927536231884, 1.05121293800539, 1.67663757954501, 2.9419480568152, 
-0.422309596621509, 2.67987715706347, 0.0249243368346056, 2.03260714794249
), SCOTLAND = c(5.15222482435597, 4.12026726057908, 5.40106951871658, 
8.67579908675796, -0.280112044817908, 2.94943820224719, 1.04592996816735, 
1.21512151215122, 1.33392618941751, 3.59806932865292, 0.974163490046604, 
0.125838926174496, 1.46627565982404, 3.42691990090835, -0.838323353293421, 
1.97262479871176, 3.40702724042636, 4.30649410147751, 2.44866586142527, 
1.93997856377279, 2.09581887638873, 4.22573890357352, 0.833278440155458, 
4.15155969296095, 2.01655899140689, 1.93980755633434, 0.325693606755129, 
0.796561260069754, -0.381713535919834, 2.90974405029185), NIRELAND = c(4.54545454545454, 
4.94752623688156, 4.42857142857145, 2.96397628818967, 6.06731620903454, 
0.0835073068893502, -1.66875260742594, -2.96987696224015, -1.18058592041975, 
-0.884955752212393, -1.74107142857143, -0.545206724216265, 1.96436729100047, 
-0.224014336917564, -1.84104176021554, 1.6010978956999, 1.42278253039172, 
1.97993429814437, 1.29287828660979, 1.61158623060724, 2.28387751649466, 
1.84005954349984, 1.79057208981284, 2.22177901874749, 2.88757950598978, 
-0.731975575530031, 3.07939176281808, -0.0593031875463392, -1.05696484201158, 
3.40717418194087), UK = c(5.76890543055322, 7.20302836425676, 
7.39543442582184, 7.22885986848197, 3.23472252213347, 2.95766398929048, 
1.20271423347285, -0.554061107319231, 0.98913965036942, 1.55113136643479, 
0.373986300291293, -1.61195434757029, 1.59052858167903, 2.07573082205217, 
1.17628969016684, 2.44680851063832, 2.84453345201007, 4.10010457610617, 
2.88208396840793, 1.58922558922557, 3.67559326527908, 3.90013106997858, 
1.36611181194425, 4.12505691303686, 2.02017257462689, 2.93167985827357, 
1.54068234183715, 2.12149379408387, 0.594313861969269, 3.83755588673622
)), row.names = c(NA, 30L), class = "data.frame")

Code:

in_sampleARDL <- data %>% 
  dplyr::filter(Date < '2020-03-01')

out_sampleARDL <-data %>% 
  dplyr::filter(Date >= '2020-03-01')

# Model Building
    
# Create the formulas

indep_vars <- expression(Industrialproduction, Householdconsumption, Investmentgrowth, ConsumerPriceIndex, Employment, Unemploymentrate, 
                         Stockmarketindex, Economicgrowth, Consumptiongrowth, Governmentexpenditure, Longtermgovernmentbondyield,
                         BankRate, ConsumerConfidenceIndex, RealPersonalDisposableIncome, PersonalDisposableIncome, SPPricechange, 
                         HouseStarts, HouseCompleted, TermSpread, BuildingPermits)

dep_vars   <- expression(NORTH, YORKSANDTHEHUMBER, NORTHWEST, EASTMIDS, WESTMIDS, EASTANGLIA, OUTERSEAST, OUTERMET, LONDON,
                         SOUTHWEST, WALES, SCOTLAND, NIRELAND, UK)

# Formulae with diff()

formulae <- unlist(lapply(dep_vars, \(x) lapply(indep_vars, \(y) bquote(.(x)~diff(.(y))))))
length(formulae)

# Without diff()

formulae2 <- unlist(lapply(dep_vars, \(x) lapply(indep_vars, \(y) bquote(.(x)~.(y)))))
length(formulae2)

result <- vector('list', length = length(formulae))
names(result) <- formulae2

y <- in_sampleARDL
ourCall2 <- "ardl(formula(formulae2[[i]]),
                      data = in_sampleARDL, max_order = 4, selection = 'BIC')"
ourValue2 <- c("pred")

for (i in seq_along(formulae)){
  
  # auto_ardl
  result[[i]][[1]] <- auto_ardl(formula(formulae2[[i]]),
                                data = in_sampleARDL, 
                                max_order = 4, selection = 'BIC')
  # prediction
  # 
  result[[i]][[2]] <- forecast(ardlDlm(formula = formula(formulae[[i]]), 
                                       #---------------extract one col------------------\/-\/-\/-
                                       data = in_sampleARDL, p = 3)$data[[i]],
                               x = out_sampleARDL |> 
                                 select(sub("\\s~.*", "", formula(formulae[[i]]))) |> 
                                 pull(), h = 4)
  # error
  result[[i]][[3]] <- mean((out_sampleARDL |> select(sub("\\s~.*",
                                                         "", formula(formulae[[i]]))) |> 
                              pull() |> (\(x) x[1:4])() - 
                              result[[i]][[2]][["forecasts"]])^2)
  # Rolling_Origin
  result[[i]][[4]] <- apply(y, 2, function(y){ return(list(ro(y, h=4, origins = 68,
                                                              call=ourCall2, value=ourValue2)))})
  # set names
  names(result[[i]]) <- c('auto_ardl','forecast','error','Rolling_Origin')
  
}

FactOREO · August 20, 2022, 5:46pm

Hello again,

so I think I've come up with a somewhat feasible solution. I added some comments in between, so you can follow what I did with the code.

I have changed the subsets, since the data you have provided doesn't reach up to 2020
I modified your call, so the greybox:ro() function can handle the output of prediction(auto_ardl()), which it could not before
You have used ardl(max_order = 4, selection = 'BIC') beforehand. This does not work since ardl() does not have such parameters. Instead I decided fitting a model via auto_ardl() with BIC as criterion, predicting upon this model and then continue.

Be aware that the calculations will need a lot of time (might consider parallelization inside ro(), see documentation). However, this is what I did:

library('tidyverse')
library('greybox')
library('ARDL')
library('dLagM')

# changed date values, since your data does not range up to 2020
in_sampleARDL <- data %>% 
  filter(Date < '1985-03-01')

out_sampleARDL <-data %>% 
  filter(Date >= '1985-03-01')

# Model Building

# Create the formulas

indep_vars <- expression(Industrialproduction, Householdconsumption, Investmentgrowth, ConsumerPriceIndex, Employment, Unemploymentrate, 
                         Stockmarketindex, Economicgrowth, Consumptiongrowth, Governmentexpenditure, Longtermgovernmentbondyield,
                         BankRate, ConsumerConfidenceIndex, RealPersonalDisposableIncome, PersonalDisposableIncome, SPPricechange, 
                         HouseStarts, HouseCompleted, TermSpread, BuildingPermits)

dep_vars   <- expression(NORTH, YORKSANDTHEHUMBER, NORTHWEST, EASTMIDS, WESTMIDS, EASTANGLIA, OUTERSEAST, OUTERMET, LONDON,
                         SOUTHWEST, WALES, SCOTLAND, NIRELAND, UK)

# Formulae with diff()

formulae <- unlist(lapply(dep_vars, \(x) lapply(indep_vars, \(y) bquote(.(x)~diff(.(y))))))
length(formulae)
#> [1] 280

# Without diff()

formulae2 <- unlist(lapply(dep_vars, \(x) lapply(indep_vars, \(y) bquote(.(x)~.(y)))))
length(formulae2)
#> [1] 280

result <- vector('list', length = length(formulae))
names(result) <- formulae2

# ardl cannot be used with 'max_order =', since it has no argument with this name
# you can fit an ARDL to the data using auto_ardl() with max_order = 4, choose the best fitting model an predict with this model
# !! you have to deparse the formulae2[[i]] language object, so auto_ardl() (and consequently ro()) can handle it inside the formula() as an actual formula !!
# !! the Call is actually a bit complicated:
#    predict(auto_ardl()) returns a vector of forecasts, so you can't use the operator $ on it
#    hence we have to grab the prediction, convert it to a data.frame and then asign the right name to the prediction column
ourCall <- "predict(auto_ardl(
                              formula = formula(deparse(formulae2[[i]])),
                              data = in_sampleARDL,
                              max_order = 4,
                              selection = 'BIC')$best_model) |>
  as.data.frame() |> `colnames<-`('pred')"

for (i in seq_along(formulae)){
  # auto_ardl
  result[[i]][[1]] <- auto_ardl(formula(formulae2[[i]]),
                                data = in_sampleARDL, 
                                max_order = 4, selection = 'BIC')
  # prediction
  # !! this is problematic, since you use i as index. But your ardlDlm-Call is not as wide as the list (35 cols vs 280 here) !!
  # !! additionally: forecast() requires a full model, not just one column of data !!
  
  # result[[i]][[2]] <- forecast(ardlDlm(formula = formula(formulae[[i]]),data = in_sampleARDL, p = 3)$data[[i]],
  #                              x = out_sampleARDL |> 
  #                                select(sub("\\s~.*", "", formula(formulae[[i]]))) |> 
  #                                pull(), h = 4)
  
  result[[i]][[2]] <- forecast(
                        ardlDlm(
                          formula = formula(formulae[[i]]),
                          data = in_sampleARDL,
                          p = 3),
                        x = out_sampleARDL |> select(sub("\\s~.*","", formula(formulae[[i]]))) |> pull(),
                        h = 4)
  # error (measured as mean squared distance between the 4 prediction values and the 4 actual values)
  result[[i]][[3]] <- mean((out_sampleARDL |> select(sub("\\s~.*","", formula(formulae[[i]]))) |> 
                              pull() |> (\(x) x[1:4])() - result[[i]][[2]][["forecasts"]])^2)
  # Rolling Origin (changed number of origins to 5)
  result[[i]][[4]] <- ro(data = in_sampleARDL |> select(sub("\\s~.*","", formula(formulae[[i]]))) |> 
                           pull(),
                         h = 5,
                         origins = 5,
                         call = ourCall,
                         value = 'pred')$pred
  
  # set names
  names(result[[i]]) <- c('auto_ardl','forecast','error','Rolling_Origin')
  
}
#> Warning: The `x` argument of `as_tibble.matrix()` must have unique column names if `.name_repair` is omitted as of tibble 2.0.0.
#> Using compatibility `.name_repair`.
#> This warning is displayed once every 8 hours.
#> Call `lifecycle::last_lifecycle_warnings()` to see where this warning was generated.

^{Created on 2022-08-20 by the reprex package (v2.0.1)}

Good luck on your further journey!

Kind regards

Edit: Had not changed the selection of the dependent variable, which should be predicted inside the ro() call. Fixed it.

system · August 27, 2022, 5:46pm

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