Hw 6

Do the exercises 9.1, 9.2, 9.3, 9.5, 9.6, 9.7, 9.8 in Hyndman.

  1. Figure 9.32 shows the ACFs for 36 random numbers, 360 random numbers and 1,000 random numbers.

a. Explain the differences among these figures. Do they all indicate that the data are white noise?

They do all indicate the data is white noise. Although it might be hard to tell on the right one since its showing series of 1,000. The main difference between these 3 is the number of samples present, with less numbers the ACF estimate will seem to have a higher variance especially in comparision to the middle and right visual.

  1. Why are the critical values at different distances from the mean of zero? Why are the autocorrelations different in each figure when they each refer to white noise?

The the blue lines, or critical values are smaller and closer to zero with more data, and wider in the first visual with less data of 36 numbers. The true autocorrelation for a white noise process is zero, so its different in each in each figure due to the change in sample size.

  1. A classic example of a non-stationary series are stock prices. Plot the daily closing prices for Amazon stock (contained in gafa_stock), along with the ACF and PACF. Explain how each plot shows that the series is non-stationary and should be differenced.

The autplot, on top, shows tha=e variance changing and that there is a trend, which makes the series non-stationary. The ACF plot is decreasing very slowly which indicates that it is non-staionary. Additionally, the PACF shows a large autocorrelation on lag 1 and then drops quickly which indicates that the series is non-stationary. In order to make a non-stationary series stationary, I can transform the series to stabilize the variance, and might still need to complete differencing to stabilize the mean. The order of differencing needed can be found using the KPSS test.

library(fpp3)
## Registered S3 method overwritten by 'tsibble':
##   method               from 
##   as_tibble.grouped_df dplyr
## ── Attaching packages ──────────────────────────────────────────── fpp3 1.0.1 ──
## ✔ tibble      3.2.1     ✔ tsibble     1.1.5
## ✔ dplyr       1.1.4     ✔ tsibbledata 0.4.1
## ✔ tidyr       1.3.1     ✔ feasts      0.4.1
## ✔ lubridate   1.9.3     ✔ fable       0.4.0
## ✔ ggplot2     3.5.1
## ── Conflicts ───────────────────────────────────────────────── fpp3_conflicts ──
## ✖ lubridate::date()    masks base::date()
## ✖ dplyr::filter()      masks stats::filter()
## ✖ tsibble::intersect() masks base::intersect()
## ✖ tsibble::interval()  masks lubridate::interval()
## ✖ dplyr::lag()         masks stats::lag()
## ✖ tsibble::setdiff()   masks base::setdiff()
## ✖ tsibble::union()     masks base::union()
gafa_stock %>% filter(Symbol == "AMZN") %>%
  gg_tsdisplay(Close, plot_type='partial') +
  labs(title = "Amazon Closing Price")
## Warning: Provided data has an irregular interval, results should be treated with caution. Computing ACF by observation.
## Provided data has an irregular interval, results should be treated with caution. Computing ACF by observation.

3. For the following series, find an appropriate Box-Cox transformation and order of differencing in order to obtain stationary data.

  1. Turkish GDP from global_economy.
  2. Accommodation takings in the state of Tasmania from aus_accommodation.
  3. Monthly sales from souvenirs.

EDIT

library(latex2exp)

turkey <- global_economy %>%
  filter(Country == "Turkey")

lambda <- global_economy %>%
  filter(Country == "Turkey") %>%
  features(GDP, features = guerrero) %>%
  pull(lambda_guerrero)

turkey_transformed <- box_cox(turkey$GDP, lambda)

turkey_tsibble <- turkey %>%
  as_tsibble(index = Year, key = Country) %>%
  mutate(Transformed_GDP = turkey_transformed)

gg_tsdisplay(turkey_tsibble, GDP, plot_type = 'partial') +
  labs(title = "Original Turkish GDP")

gg_tsdisplay(turkey_tsibble, plot_type = 'partial') +
  labs(title = TeX(paste0("Transformed Turkish GDP with $\\lambda$ = ",
                           round(lambda, 2))))
## Plot variable not specified, automatically selected `y = GDP`

#still needs to be differenced
library(tseries)
## Registered S3 method overwritten by 'quantmod':
##   method            from
##   as.zoo.data.frame zoo
turkey_tsibble %>%
    features(Transformed_GDP, unitroot_ndiffs)
## # A tibble: 1 × 2
##   Country ndiffs
##   <fct>    <int>
## 1 Turkey       1
turkey_tsibble %>%
  mutate(Differenced_GDP = Transformed_GDP %>% difference(1)) %>%
  gg_tsdisplay(Differenced_GDP, plot_type = 'partial')
## Warning: Removed 1 row containing missing values or values outside the scale range
## (`geom_line()`).
## Warning: Removed 1 row containing missing values or values outside the scale range
## (`geom_point()`).

Tasmania <- aus_accommodation %>%
  filter(State == "Tasmania") 

gg_tsdisplay(Tasmania, Takings, plot_type = 'partial') +
  labs(title = "Original Tasmania Takings")

Tasmania %>%
  features(Takings, unitroot_ndiffs)
## # A tibble: 1 × 2
##   State    ndiffs
##   <chr>     <int>
## 1 Tasmania      1
Tasmania %>%
  features(Takings, unitroot_nsdiffs)
## # A tibble: 1 × 2
##   State    nsdiffs
##   <chr>      <int>
## 1 Tasmania       1
lambda <- Tasmania %>%
  features(Takings, features = guerrero) %>%
  pull(lambda_guerrero)

t_transformed <- box_cox(Tasmania$Takings, lambda)

# lag 4 = seasonal
Tasmania %>% gg_tsdisplay(difference(t_transformed, 4), plot_type='partial') +
  labs(title = TeX(paste0("Seasonal Differenced Tasmania with $\\lambda$ = ",
         round(lambda,2))))
## Warning: Removed 4 rows containing missing values or values outside the scale range
## (`geom_line()`).
## Warning: Removed 4 rows containing missing values or values outside the scale range
## (`geom_point()`).

#still needs one more order of difference 
Tasmania %>%
  gg_tsdisplay(difference(difference(t_transformed, 4)), plot_type = 'partial') +
  labs(title = TeX(paste0("Seasonal + one order Differenced Tasmania with $\\lambda$ = ", round(lambda, 2))))
## Warning: Removed 5 rows containing missing values or values outside the scale range
## (`geom_line()`).
## Warning: Removed 5 rows containing missing values or values outside the scale range
## (`geom_point()`).

** does it matter if unitroot_nsdiffs if done before or after transformation? ** internet says after transformation

souvenirs %>%
  gg_tsdisplay(Sales, plot_type='partial', lag = 36) +
  labs(title = "Non-transformed Monthly Souvenir Sales")

lambda <- souvenirs %>%
  features(Sales, features = guerrero) %>%
  pull(lambda_guerrero)

souvenirs %>%
  features(box_cox(Sales,lambda), unitroot_nsdiffs) #one seasonal differencing
## # A tibble: 1 × 1
##   nsdiffs
##     <int>
## 1       1
souvenirs %>%
  features(box_cox(Sales,lambda), unitroot_ndiffs) #one differencing
## # A tibble: 1 × 1
##   ndiffs
##    <int>
## 1      1
souvenirs %>%
  gg_tsdisplay(difference(difference(box_cox(Sales,lambda), 12)), plot_type='partial', lag = 36) +
  labs(title = TeX(paste0("Differenced Monthly Souvenir Sales with $\\lambda$ = ",
         round(lambda,2))))
## Warning: Removed 13 rows containing missing values or values outside the scale range
## (`geom_line()`).
## Warning: Removed 13 rows containing missing values or values outside the scale range
## (`geom_point()`).

  1. For your retail data (from Exercise 7 in Section 2.10), find the appropriate order of differencing (after transformation if necessary) to obtain stationary data
set.seed(12345678)
myseries <- aus_retail |>
  filter(`Series ID` == sample(aus_retail$`Series ID`,1)) #monthly

myseries %>%
  gg_tsdisplay(Turnover, plot_type='partial', lag = 36) +
  labs(title = "Non-transformed Retail")

lambda <- myseries %>%
  features(Turnover, features = guerrero) %>%
  pull(lambda_guerrero)

myseries <- myseries %>%
  mutate(Transformed_Turnover = box_cox(Turnover, lambda))

myseries %>%
  features(Transformed_Turnover, unitroot_ndiffs)
## # A tibble: 1 × 3
##   State              Industry                                            ndiffs
##   <chr>              <chr>                                                <int>
## 1 Northern Territory Clothing, footwear and personal accessory retailing      1
myseries %>%
  features(Transformed_Turnover, unitroot_nsdiffs)
## # A tibble: 1 × 3
##   State              Industry                                            nsdiffs
##   <chr>              <chr>                                                 <int>
## 1 Northern Territory Clothing, footwear and personal accessory retailing       1
myseries %>%
  gg_tsdisplay(difference(difference(box_cox(Turnover,lambda), 12)), plot_type='partial', lag = 36) +
  labs(title = TeX(paste0("Differenced Monthly Souvenir Sales with $\\lambda$ = ",
         round(lambda,2))))
## Warning: Removed 13 rows containing missing values or values outside the scale range
## (`geom_line()`).
## Warning: Removed 13 rows containing missing values or values outside the scale range
## (`geom_point()`).

6. Simulate and plot some data from simple ARIMA models.

  1. Use the following R code to generate data from an AR(1) model with with 𝜙1=0.6 and 𝜎^2=1. The process starts with 𝑦1=0
  • AR(1) = autoregressive of order 1; depends on its previous value
  • 𝜎^2 = variance, value of 1 means not white noise
  • 𝜙1=0.6 - phi - parameter of AR, strong dependence (when its 0 then its white noise)
y <- numeric(100)
e <- rnorm(100)
for(i in 2:100)
  y[i] <- 0.6*y[i-1] + e[i]
sim <- tsibble(idx = seq_len(100), y = y, index = idx)
  1. Produce a time plot for the series. How does the plot change as you change ϕ1

Changing the parameter changes the patterns in the series. Value are required to be under 1 to be considered stationary. Negative values of phi cause the series to oscillates around zero, on negative and positive values. Plot with phi of 0.9 shows a more clear trend; it is close to 1 which is typically random walk. Phi of 0.1 is closes to white noise, this plot has more fluctuations and change quickly from previous value. When phi = 0.6 it is more moderate compared to .1 and more volatile compared to .9.

set.seed(123) 
sim %>% autoplot(.vars=y)+
  labs(title = "AR(1) phi = 0.6")

for(i in 2:100)
  y[i] <- 0.1*y[i-1] + e[i]
sim1 <- tsibble(idx = seq_len(100), y = y, index = idx)
sim1 %>% autoplot(.vars=y)+
  labs(title = "AR(1) phi = 0.1")

for(i in 2:100)
  y[i] <- 0.9*y[i-1] + e[i]
sim2 <- tsibble(idx = seq_len(100), y = y, index = idx)
sim2 %>% autoplot(.vars=y)+
  labs(title = "AR(1) phi = 0.9")

for(i in 2:100)
  y[i] <- -0.6*y[i-1] + e[i]
sim3 <- tsibble(idx = seq_len(100), y = y, index = idx)
sim3 %>% autoplot(.vars=y)+
  labs(title = "AR(1) phi = -.6")

  1. Write your own code to generate data from an MA(1) model with 𝜃1=0.6 and 𝜎2=1
  • pheta = moving averge
e <- rnorm(100, mean = 0, sd = 1)  #variance =1

# MA(1) 
for (i in 2:100) {
  y[i] <- e[i] + 0.6 * e[i - 1]  # θ1 = 0.6
}

sim4 <- tsibble(idx = seq_len(100), y = y, index = idx)
sim4 %>% autoplot(.vars=y)+
  labs(title = "MA(1) pheta = 0.6")

  1. Produce a time plot for the series. How does the plot change as you change 𝜃1

Changing pheta has almost the opposite effect as phi; the plot with 0.8 has more fluctuations as the previous error terms has more influence. When pheta is .2 the plot has more fluctuations and changes quickly. Similarly negative value of pheta also has an oscillatory pattern as seen in the plot of pheta = -.5.

for (i in 2:100) {
  y[i] <- e[i] + 0.2 * e[i - 1]  }

sim5 <- tsibble(idx = seq_len(100), y = y, index = idx)
sim5 %>% autoplot(.vars=y)+
  labs(title = "MA(1) pheta = 0.2")

for (i in 2:100) {
  y[i] <- e[i] + 0.8 * e[i - 1]  }

sim6 <- tsibble(idx = seq_len(100), y = y, index = idx)
sim6 %>% autoplot(.vars=y)+
  labs(title = "MA(1) pheta = 0.8")

for (i in 2:100) {
  y[i] <- e[i] + -0.5 * e[i - 1]  }
sim7 <- tsibble(idx = seq_len(100), y = y, index = idx)
sim7 %>% autoplot(.vars=y)+
  labs(title = "MA(1) pheta = -.5")

e. Generate data from an ARMA(1,1) model with 𝜙1=0.6, 𝜃1=0.6, and 𝜎^2=1.

  • no I or difference
arma11 <- function(phi, theta, n = 100, sigma = 1) {
  e <- rnorm(n, mean = 0, sd = sigma) 
  y <- numeric(n) 
  for (i in 2:n) {
    y[i] <- phi * y[i - 1] + e[i] + theta * e[i - 1]  
  }
  tsibble(idx = seq_len(n), y = y, index = idx)
}

phi1 <- 0.6
theta1 <- 0.6
arma_series <- arma11(phi1, theta1)
  1. Generate data from an AR(2) model with 𝜙1=−0.8, 𝜙2=0.3, and 𝜎^2=1. (Note that these parameters will give a non-stationary series.)
ar2 <- function(phi1, phi2, n = 100, sigma = 1) {
  e <- rnorm(n, mean = 0, sd = sigma)  
  y <- numeric(n)                      
  
  for (i in 3:n) {                     # 2 phetas means start at 3rd
    y[i] <- phi1 * y[i - 1] + phi2 * y[i - 2] + e[i]  
  }
  
  tsibble(idx = seq_len(n), y = y, index = idx)
}

phi1 <- -0.8
phi2 <- 0.3

ar2_series <- ar2(phi1, phi2)
  1. Graph the latter two series and compare them.

ARMA (1,1) has residuals that appear to be white noise; the autoplot look like its stationary, with no trends, and fluctuates between 4 and -2. The AR(2) plot has data on zero about half the plot then it begins to exponentially increases and decreases. AR(2) does not appear to be stationary, ACF doesnt show white noise and oscillates, the PACF only has one lag present. Something seems off and this is because ϕ2-ϕ1 =0.3−(−0.8) = 1.1 and is supposed to be under 1 for AR to be stationary.

arma_series%>%
  gg_tsdisplay(y, plot_type='partial') +
  labs(title = TeX("ARMA(1,1) with ϕ1 = 0.6 and θ1 = 0.6"))

ar2_series %>%
  gg_tsdisplay(y, plot_type='partial') +
  labs(title = TeX("AR(2)  with ϕ1 = -0.8 and ϕ2 = 0.3"))

  1. Consider aus_airpassengers, the total number of passengers (in millions) from Australian air carriers for the period 1970-2011.
  1. Use ARIMA() to find an appropriate ARIMA model. What model was selected. Check that the residuals look like white noise. Plot forecasts for the next 10 periods.

The ARIMA (0,2,1) was selected. The residuals in both the ACF and PACF look like white noise.

fit <- aus_airpassengers |>
  filter(Year < 2008) %>%
  model(ARIMA(Passengers))

report(fit)
## Series: Passengers 
## Model: ARIMA(0,2,1) 
## 
## Coefficients:
##           ma1
##       -0.8810
## s.e.   0.0795
## 
## sigma^2 estimated as 2.464:  log likelihood=-67.55
## AIC=139.1   AICc=139.47   BIC=142.27
residuals <- fit %>% residuals()
gg_tsdisplay(residuals, plot_type = "partial") + 
  labs(title = "Residuals of the ARIMA Model")
## Plot variable not specified, automatically selected `y = .resid`

fit %>% 
  forecast(h=10) %>%
  autoplot(aus_airpassengers)

b. Write the model in terms of the backshift operator.

\[ y_t = -0.87 \epsilon_{t-1} + \epsilon_t \]

\[ (1 - B)^2 y_t = (1 - 0.87 B) \epsilon_t \]

  1. Plot forecasts from an ARIMA(0,1,0) model with drift and compare these to part a.

Model in a: ARIMA(2,1,0) has a better looking forecast than the ARIMA (0,1,0) in c, which looks like the forecasts fall below the actual points, or the prediction interval is too narrow. The ACF and PACF residuals similar but ARIMA(2,1,0) has more negative lags and this model ARIMA (0,1,0) has more positive lags.

fit2 <-aus_airpassengers %>%
  filter(Year < 2008) %>%
  model(ARIMA(Passengers ~ pdq(0,1,0) +1 )) # Passengers ~ drift() or +1

fit2 %>% 
  forecast(h=15) %>%
  autoplot(aus_airpassengers) +
  labs(title = "Australian Passengers with ARIMA(0,1,0)", y = "Passengers (in millions)")

residuals <- fit2 %>% residuals()
gg_tsdisplay(residuals, plot_type = "partial") + 
  labs(title = "Residuals for ARIMA(0,1,0) + drift")
## Plot variable not specified, automatically selected `y = .resid`

  1. Plot forecasts from an ARIMA(2,1,2) model with drift and compare these to parts a and c. Remove the constant and see what happens.

The lags in this model are much shorter than a and c. The residuals also suggest this model is stationary. When removing the constant, errors come up and then residuals appear to have NA values preventing plotting. Removing the constant has many potential impact- on the mean, and for example, removing the constant results in a trend of order 0.

fit3 <-aus_airpassengers %>%
  filter(Year < 2008) %>%
  model(ARIMA(Passengers ~ pdq(2,1,2) +1 )) # Passengers ~ drift() or +1

fit3 %>% 
  forecast(h=15) %>%
  autoplot(aus_airpassengers) +
  labs(title = "Australian Passengers with ARIMA(2,1,2)", y = "Passengers (in millions)")

residuals <- fit3 %>% residuals()
gg_tsdisplay(residuals, plot_type = "partial") + 
  labs(title = "Residuals for ARIMA(2,1,2) + drift")
## Plot variable not specified, automatically selected `y = .resid`

sum(is.na(aus_airpassengers))
## [1] 0
fit3 <- aus_airpassengers %>%
  filter(Year < 2008) %>%
  model(ARIMA(Passengers ~ pdq(2,1,2) + 0))  # Using +0 instead of +1
## Warning: 1 error encountered for ARIMA(Passengers ~ pdq(2, 1, 2) + 0)
## [1] non-stationary AR part from CSS
residuals <- fit3 %>% residuals()

sum(is.na(residuals))
## [1] 38
if (all(!is.na(residuals))) {
  gg_tsdisplay(residuals, plot_type = "partial") + 
    labs(title = "Residuals for ARIMA(2,1,2) without constant")
} else {
  message("Residuals contain NA values; cannot plot.")
}
## Residuals contain NA values; cannot plot.
  1. Plot forecasts from an ARIMA(0,2,1) model with a constant. What happens?

There is a warning: “Model specification induces a quadratic or higher order polynomial trend”.The Forecast looks like a better match than some of the other models. The slope also changes a bit. This is the auto model R gave us in a but with the constant added.

fit3 <-aus_airpassengers %>%
  filter(Year < 2008) %>%
  model(ARIMA(Passengers ~ pdq(0,2,1) +1 )) # Passengers ~ drift() or +1
## Warning: Model specification induces a quadratic or higher order polynomial trend. 
## This is generally discouraged, consider removing the constant or reducing the number of differences.
fit3 %>% 
  forecast(h=15) %>%
  autoplot(aus_airpassengers) +
  labs(title = "Australian Passengers with ARIMA(0,2,1))", y = "Passengers (in millions)")

residuals <- fit3 %>% residuals()
gg_tsdisplay(residuals, plot_type = "partial") + 
  labs(title = "Residuals for ARIMA(0,2,1)) + drift")
## Plot variable not specified, automatically selected `y = .resid`

  1. For the United States GDP series (from global_economy):
us_gdp <- global_economy %>% filter(Country == "United States")

gg_tsdisplay(us_gdp, plot_type = "partial")  #not stationary
## Plot variable not specified, automatically selected `y = GDP`

a. if necessary, find a suitable Box-Cox transformation for the data;

lambda <-us_gdp %>%
  features(GDP, features = guerrero) %>%
  pull(lambda_guerrero)

us_gdp_transformed <- us_gdp %>%
  mutate(Transformed_GDP = box_cox(GDP, lambda))
  1. fit a suitable ARIMA model to the transformed data using ARIMA();
us_gdp_transformed %>%
  features(Transformed_GDP, unitroot_ndiffs)
## # A tibble: 1 × 2
##   Country       ndiffs
##   <fct>          <int>
## 1 United States      1
us_gdp_transformed <- us_gdp_transformed %>%
  mutate(Differenced_GDP = difference(Transformed_GDP))

best_fit <- us_gdp_transformed %>%
  model(auto_arima = ARIMA(Differenced_GDP))

report(best_fit)
## Series: Differenced_GDP 
## Model: ARIMA(1,0,0) w/ mean 
## 
## Coefficients:
##          ar1  constant
##       0.4586  118.1789
## s.e.  0.1198    9.5047
## 
## sigma^2 estimated as 5381:  log likelihood=-325.32
## AIC=656.65   AICc=657.09   BIC=662.83
  1. try some other plausible models by experimenting with the orders chosen;
usa_fit <- global_economy %>%
  filter(Country == "United States") %>%
  model(arima120 = ARIMA(box_cox(GDP,lambda) ~ pdq(1,2,0)),
        arima210 = ARIMA(box_cox(GDP,lambda) ~ pdq(2,1,0)),
        arima111 = ARIMA(box_cox(GDP,lambda) ~ pdq(1,1,1)))

glance(usa_fit) %>% arrange(AICc) %>% select(.model:BIC)
## # A tibble: 3 × 6
##   .model   sigma2 log_lik   AIC  AICc   BIC
##   <chr>     <dbl>   <dbl> <dbl> <dbl> <dbl>
## 1 arima120  6780.   -326.  656.  656.  660.
## 2 arima111  5580.   -325.  659.  659.  667.
## 3 arima210  5580.   -325.  659.  659.  667.
  1. choose what you think is the best model and check the residual diagnostics;
# the AICc value of a model from c, plausible models, is lower than the autoplot
usa_fit %>%
  select(arima120) %>%
  gg_tsresiduals() +
  ggtitle("Residuals for ARIMA(1,2,0)")

#residuals prove to be white noise and non
  1. produce forecasts of your fitted model. Do the forecasts look reasonable?
usa_fit <- us_gdp_transformed %>%
  model(arima120 = ARIMA(box_cox(GDP, lambda) ~ pdq(1, 2, 0)))

forecasts <- usa_fit %>%
  forecast(h = 8)

autoplot(forecasts, us_gdp_transformed) +
  ggtitle("Forecast for US GDP with ARIMA(1,2,0)") +
  labs(y = "GDP (Transformed)", x = "Year")

accuracy(usa_fit)
## # A tibble: 1 × 11
##   Country    .model .type      ME    RMSE     MAE   MPE  MAPE  MASE RMSSE   ACF1
##   <fct>      <chr>  <chr>   <dbl>   <dbl>   <dbl> <dbl> <dbl> <dbl> <dbl>  <dbl>
## 1 United St… arima… Trai… -1.84e9 1.74e11 1.05e11 0.130  1.64 0.308 0.427 0.0397
  1. compare the results with what you would obtain using ETS() (with no transformation).

Visually the forecast for the ARIMA model has a larger slope than ETS. The AICc cannot be be used to compare the models since ARIMA has differencing and ETS does not. The RMSE value is needed. However, I’m unsure why the RMSE values for both models are so larger, I may have made an error. Actually, I did not invert the scale bak after transfroming the data for the AIRMA model but that does not explain it doe ETS since there was no transformation. Based on these results though the ETS model has the lower RMSE value. There is a lot more improvements I can make but I encounter more errors as I tried. So i will just submit this and hope for the best.

ets_fit <- us_gdp %>%
  model(ETS_Model = ETS(GDP))

ets_forecasts <- ets_fit %>%
  forecast(h = 8)

autoplot(us_gdp, GDP) +
  autolayer(ets_forecasts, series = "ETS Forecasts") +
  ggtitle("ETS Forecasts for US GDP")
## Warning in ggdist::geom_lineribbon(without(intvl_mapping, "colour_ramp"), :
## Ignoring unknown parameters: `series`
## Warning in geom_line(mapping = without(mapping, "shape"), data =
## unpack_data(object[single_row[["FALSE"]], : Ignoring unknown parameters:
## `series`

accuracy(ets_fit)
## # A tibble: 1 × 11
##   Country     .model .type      ME    RMSE     MAE   MPE  MAPE  MASE RMSSE  ACF1
##   <fct>       <chr>  <chr>   <dbl>   <dbl>   <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 United Sta… ETS_M… Trai… 2.10e10 1.67e11 1.05e11 0.484  1.91 0.307 0.409 0.153