[SOLVED] Indr 450/550 homework 1

1. Forecasting Coffee Prices (monthly average price index of coffee in
commodity markets as reported by the IMF commodity price data
portal)
(a) Plot the data and visually assess whether there is significant trend
and seasonality.
(b) To obtain a benchmark for errors, implement the naive one-month
ahead forecast i) ˆyt = yt−1. Report the Mean Absolute Error (MAE), Mean Absolute Percentage Error (MAPE) and Root
Mean Squared Error (RMSE) of the naive forecast for years 1991
through 2020. These error measures constitute a simple benchmark for all other approaches (i.e. hopefully you will obtain lower
errors by more sophisticated methods). There is plenty of data
in this case and we can start forecasting from 1990 but different
methods require different initializations in the first year so it’s
best to compare the errors from 1991 to the end of 2020 and we
use 2021 for some basic testing.
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(c) Use a 5-period moving average to forecast the one month-ahead
price. Report the MAE, MAPE and RMSE of the forecast for
years 1991 through 2020. Report 95 percent prediction intervals
(using the RMSE estimated in years 1991 to 2020) for the onemonth ahead forecasts for year 2021. How do your prediction
intervals perform for 2021?
(d) Use exponential smoothing to forecast the one-month ahead price.
Perform an exhaustive search for the best smoothing constant α
(that leads to the minimum MSE). Report the MAE, MAPE
and RMSE of the forecast for years 1991 through 2020. Report
95 percent prediction intervals (using the RMSE estimated in
years 1991 through 2020) for the one-month ahead forecasts for
2021. How do these compare with the benchmark and the MA-5
forecast?
(e) The data seems to have trend. Implement a naive forecast that
includes trend: ˆyt = yt−1 + (yt−2 − yt−3) in order to forecast the
one-month ahead price. Report the MAE, MAPE and RMSE
of the forecast for years 1991 through 2020. Report 95 percent
prediction intervals (using the RMSE estimated in years 1991
through 2020) for the one-month ahead forecasts for 2021. How
do these compare with the previous forecasts?
(f) Implement an exponentially smoothed version of the previous
forecast (you can take α = 0.7 and β = 0.2): ˆy = αyt−1 + (1 −
α)ˆyt−1+βzt−1+(1−β)ˆzt−1 where zt = yt−yt−1 and ˆzt = ˆyt−yˆt−1
for one-month ahead prices. Report the MAE, MAPE and RMSE
of the forecast for years 1991 through 2020. Report 95 percent
prediction intervals (using the RMSE estimated in years 1991
through 2020) for the one-month ahead forecasts for 2021. . How
do these compare with the previous forecasts?
(g) Find the values of α and β that minimize the RMSE for sixmonth ahead forecasts for years 1991 through 2020. For the
optimal values of the parameters, predict the mean coffee prices
for the next six months from January 2021 to June 2022. Note
that ˆyt+k = αyt−1 + (1 − α)ˆyt−1 + (k + 1)(βzt−1 + (1 − β)ˆzt−1).
Report the MAE, MAPE and RMSE of the six-month ahead
forecast for years 1991 through 2020.
(h) Your report for the exercise should include a table similar to the
one below.
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Method Spec. RMSE MAPE
Benchmark –
MA-5 –
ES –
Trend –
Smoothed Trend
6-month ahead
Note that the model specification for exponential smoothing is:
α
∗ = …, β
∗ = … .
2. Generate 500 realizations from an AR-1 process Yt = c + ϕ1Yt−1 + ϵt
where c = 50, ϕ1 = 0.6 and ϵt are normally distributed with mean zero
and standard deviation 20.
(a) Plot the generated observations.
(b) Plot the Auto-Correlation Function (start from observation 100
to eliminate the effects of initialization).
(c) Implement a naive forecast on the data ˆyt = yt−1. Compute the
RMSE starting from period 100.
3. Investigating the Auto-correlation structure of the Coffee Price Index
Data.
(a) Plot the ACF and PACF of the coffee price index time series.
State if there are interesting observations.
(b) Difference the data: ut = yt − yt−1. Plot the ACF and PACF of
the differences ut
. State if there are interesting observations.
(c) Based on the previous AC analysis, are you able to propose a
model for the original data series?