Explore the Properties of the Consumer Price Index - MATLAB Assignment Help

Assignment Task:

Task:

1) In this problem you explore the properties of the consumer price index and unemployment rate series from Italy1and select and evaluate different time series models.

a) The file cpi it.csv contains data on seasonally non-adjusted consumer price in-dex (CPI) for Italy from 1985Q1 - 2019Q4. Compute and plot the log, the annualizedquarter-over-quarter growth rate and the annual growth rate for the CPI of Italy. De-scribe the main features of the time series briefly using words. Investigate the order of integration for log CPI series using appropriate Augmented Dickey Fuller (ADF) tests with 3 lagged differences. Motivate your choice of deterministic terms. What is your conclusion from the unit root analysis? 1Data source: FRED.

b) Now consider the file unrate it.csv. It contains data on seasonally adjusted un- employment rate in % for Italy from 1985Q1 - 2019Q4. Plot and briefly describe the unemployment series for Italy (yt). Conduct a KPSS test on yt

. Explain your speci-fication choice for the test. What do you conclude from your result? Is this result in line with a unit root analysis based on an ADF test and a Phillips-Perron test for yt?
Explain.2
Use yt (the level of the unemployment rate) in the following.
c) Plot the SACFs and SPACFs for yt. Based on these graphs, suggest a pure AR. Explain your choice briefly.

d) Estimate an AR(3) model with intercept for yt using all available data and the (condi-tional) LS method. Report the parameter estimates together with the asymptotic stan- dard errors. Conduct a Chow breakpoint and Chow forecast test to test for a parameter change after 2007Q2. What do you conclude from your test results?
e) Apply the information criteria discussed in the lecture to select a pure AR model for yt (use models with an intercept, pmax = 8). What are the suggested lag orders?

f) Estimate the models suggested in 1e). For each model, provide plots of standardized residuals and residual ACFs and PACFs. Conduct diagnostic tests for remaining resid-ual autocorrelation, ARCH effects and non-normality for each of the estimated mod-els. Describe your results and comment on the adequacy of each model. Which model would you prefer?

g) A researcher includes in the AR(3) model an impulse dummy to capture an unusually large residual in 1994Q4. The dummy variable takes on value 1 in 1994Q4 and is zero elsewhere. Report the estimation results and repeat the test for non-normality. How does your result change (compared to the one in 1f)?

h) Now you want to estimate the AR(3) model for yt using exact ML. Report the MLE estimates of c, α1, α2, α3, σ2 together with their asymptotic standard errors. Note: Forthis part you have to program the ML estimator on your own. This involves setting up the exact log-likelihood function and using a Matlab optimizer.3 What is the value of the maximized log-likelihood function? Which of the coefficients are significantly different from zero?

2) In this problem you analyze the properties of unit root tests by a Monte Carlo simulation. You may use the existing Matlab functions for both unit root tests.
a) Consider the data generating process yt = δt + c + εt , t = 1, . . . , T, εt iid∼ N(0, 1).
i. Explain why simulating from this process can be thought of as simulating a time series under the H0 of a KPSS test.
ii. Generate M = 2000 sets of time series of length T = 5000 using c = 0 and δ = 0. For each time series, conduct a KPSS test with a constant and lag truncation l12 = [12(T/100)1/4 ] and store the value of the test statistic. Report the 0.9, 0.95, and 0.99 quantiles of the test statistic distribution.
iii. Generate M = 2000 sets of time series of length T = 5000 using c = 0 and δ = 0.1. For each time series, conduct a KPSS test with a trend and lag truncation l12 = [12(T/100)1/4 ] and store the value of the test statistic. Report the 0.9, 0.95, and 0.99 quantiles of the test statistic distribution.

2For the KPSS and the Phillips-Perron test use the data-driven lag truncation choice discussed in the lecture. 3For this part, the Matlab arima/estimate functions should only be used for checking your results.

iv. Relate the reported quantiles to the asymptotic critical values of the KPSS test.
b) Repeat problems 2c) to 2f) for the following values of α1, α2 and T
(α1, α2) ∈ {(0.4, 0.2),(0.5, 0.2),(0.6, 0.2),(0.7, 0.2), ,(0.8, 0.2)} T ∈ {50, 100, 200, 500}

c) Simulate for a given α1, α2 and T, M = 2000 sets of time series from the DGP yt = (1 − α1 − α2)μ0 + α1yt−1 + α2yt−2 + εt

, (1)where y−1 = y0 = 0, μ0 = 2 and εt iid∼ N(0, 1).

d) For each of your generated M time series, conduct a KPSS test with a constant at the 5% level. Use the lag truncation parameter lq from the lecture. For both q = 4 and q = 12, record how often the H0 is rejected in the M replications and compute the empirical rejection frequency (number of rejections/M). Explain which of the reported values correspond to the empirical size and power of the test. What size values would you expect in an ideal test?

e) For each of your generated M time series, conduct an Augmented Dickey Fuller test at the 5% level with a constant (Case 2) and one lagged difference. Record how often the unit root hypothesis is rejected in the M replications and compute the empirical rejection frequency (number of rejections/M). Explain which of the reported values correspond to the empirical size and power of the test.

f) For each of your generated M time series, conduct an Augmented Dickey Fuller test at the 5% level using the test regression ?yt = c + φyt−1 +
Pp−1 i=1 α∗i ?yt−i + εt, where the number of lagged differences p − 1 is determined using the information criterion BIC. Allow between 0 and 4 lagged differences. For T = 500 report how often each lag length has been selected. Record how often the unit root hypothesis is rejected in the M replications and compute the empirical rejection frequency (number of rejections/M).

g) For the results of the KPSS tests in 2d), provide a plot for both q = 4, q = 12 andfor each T that shows for different α1 values on the x-axis the corresponding rejection frequencies on the y-axis. Explain the main differences in the rejection frequencies between the parameters q = 4 and q = 12. Which variant (l4 or l12) performs better?

h) For the results of the ADF-tests in 2e) and 2f), provide a plot for each T that shows for different α1 values on the x-axis the corresponding rejection frequencies on the y-axis.

Explain any differences between the lag order specifications.

i) Briefly describe any differences between the ADF-test and the KPSS test in terms of correct unit root detection.

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