#Please Answer in Python Codeblock

Question 3 - Model Estimation

Part (a)

Plot the auto-correlation function (ACF) to get an indication of what type of ARIMA model may apply to this dataset. Refer to the table below to determine which model is best.

Provide your model choice either in a text cell below or as comments in the code cell alongside your code for the ACF plot.

Shape	Indicated Model
Exponential, decay to zero	AR - Use PACF plot to identify order
Alternating positive and negative, decaying to zero	AR - Use PACF plot to identify order
One or more spikes, rest are essentially zero	MA - Order identified by where plot becomes zero
Decay, starting after a few lags	ARMA
All zero or close to zero	Data is essentially random
High values at fixed intervals	Include seasonal autoregressive term / difference
No decay to zero	The series is not stationary

Part (b)

Plot the partial auto-correlation function (PACF). Based on the PACF plot and ACF plot, determine what values are likely best for p and q.

Question 2 Part (c) should have determined the best value for d. Fill in your answer for p, d, and q in this text cell:

p= YOUR_ANSWER_WITH_REASON

d= YOUR_ANSWER (reason provided in Question 2 Part (c))

q= YOUR_ANSWER_WITH_REASON

C) Given your determination of p, d, and q, fit an ARIMA model with those parameters using the entire time series as training data. Get the predicted values from the model (calling predict with d as the start value and including typ='levels' if d>0) and compute the mean squared error (MSE) between the predictions and the actual values.

In addition to printing the resulting MSE, plot the predictions alongside the actual values.

D)

To further assess the accuracy of the model, plot the residual errors of the model both as a histogram and as a scatterplot.

Based on these plots, are the residuals normally distributed? Are the mean and variance constant over time? Provide reasons with your answers. Include your answer either as a text cell or in-line comment.

E) Since the estimated parameters are not guaranteed to provide the best results, try fitting an ARIMA model with new parameters (different value for at least one of p, d, and q). Based on MSE, is your new model better or worse than your original in part (a)?