Research Project: PPP - Testing and Forecasting (Due: November 5)
You will test the PPP Theory for one currency. You will check the regression used to test PPP. You will forecast that exchange rate using PPP. (You are assigned a currency according to your student number. Check your assigned currency . If you do not do the assignment using your assigned currency, you'll receive zero credit.)

1. PPP Model
The rate of change in (index) prices should be similar when measured in a common currency. (As long as trade frictions are unchanged).

To test this theory, you will use the linearized Relative PPP model. Under this model we have:  ef,t = (IDC,t - IFC,t),

where ef,t is the percentage change in exchange rates, IDC,t is the domestic inflation rate and IFC,t is the foreign inflation rate, all at time t.

1.A PPP Model: Testing
You will test Relative PPP using a regression based on a Relative PPP model:   ef,t = α + β (IDC,t - IFC,t) + εt

Under Relative PPP, the hypothesis to test is:   H0: α=0 and β=1.

1.B PPP-Based Model: Forecasting
You will forecast St+1. That is, Et[St+1] = St * (1 + Et[ef,t+1])

To get Et[ef,t+1], you will use the linearized Relative PPP based model:   Et[ef,t+1] = α + β (Et[IDC,t+1] - Et[IFC,t+1]).

To get forecasts for the inflation rate differential -i.e., Et[IDC,t+1] - Et[IFC,t+1]-, assume an AR(1) model for inflation rate differentials.


2. Data
You have monthly Consumer Price Indexes (CPI) and exchange rates (St) from January 1973 till now for 12 countries + the U.S. (For some countries, the starting date is later.) The countries are UK, Switzerland, Denmark, Norway, USA, Canada, Mexico, Japan, India, Korea, Brazil, Singapore & South Africa. Download PPP data set (ppp_m.csv) from my homepage.

2.A Preliminary Work: Transform the Data
CPI and Exchange Rates are in levels. Thus, you need to put the data into percentage changes (or log changes) to test relative PPP. That is, you need to create ef,t, IFC,t, and IDC,t.


3. Things to Do
  • Testing Model Adequacy and PPP
    (A) Visual Test
        A.1 Plot ef,t against (IDC,t - IFC,t). Check if the plot forms a 45 degree line -i.e., the PPP line.
    (B) Formal Tests
        B.1 Using the whole sample, run a linear regression. Dependent variable: ef,t. Independent variable: (IDC,t - IFC,t) . That is,
                ef,t = α + β (IDC,t - IFC,t) + εt
          Null hypothesis to test: H0:α=0 and β=1.
        B.2 Check for outliers. Plot standardized residuals and Cook's D measure. Use the rules of thumb for standardized residuals, leverage observations and Cook's D to determine if there are outliers.
        B.3 Check if there is a structural change, using a QLR test.
        B.4 Check if the errors follow a normal distribution. Perform a JB test using a 5% significance level.
        B.5 Check if the errors are homoscedastic, by doing LM-White and GQ tests, using a 5% significance level.
        B.6 Check if the errors are homoscedastic, by doing LB test with 4 lags and 12 lags for squared residuals.
        B.7 Check if the errors are serially correlated, by doing LM BG tests with 4 lags and 12 lags, using a 5% significance level.
        B.8 Check if the errors are serially correlated, by doing a Durbin Watson test. Overall, is there autocorrelation in the errors?
        B.9 Test PPP. Null hypothesis to test: α=0 and β=1. Run individual t-tests and a joint F-test. Based on your findings from B.4, use an appropriate distribution (exact or asymptotic?) for your tests.
        B.10 Do the individual tests results change if you use Newey-West S.E.?
        B.11 Compare the model's in sample MSE (= RSS/T) with a RW's in sample MSE. Which model is better?
        B.12 In the database, you also have other data: the 5 Fama-French factors (Mkt_RF, SMB, HML, RMW, & CMA), a USD Index (TWEX_BROAD), Oil prices (CRUDE_WTI), Gold prices (GOLD), and the risk free rate (RF). Transform all the nominal variables (USD Index, Oil prices, and Gold prices) into log changes. Expand the PPP model with all these additional variables. What are the drivers of changes in the exchange rate?
        B.13. Does your answer in B.12 change if you use Newey-West S.E.? Keep the inflation rate differential and all the other significant variables in your model to forecast exchange rates.
        B.14 If your model in B.13 is different from your PPP-based model, compare the augmented model's in sample MSE (= RSS/T) with a RW's in sample MSE. Which model is better?


  • Forecasting Exchange Rates
    Steps:
      1) Using the estimation period from start of your sample to December 2019, estimate the model you got from B.13. Get the estimated coefficients. Report the regression
      2) Using the estimation period from start of your sample to December 2019, run an AR(1) regression for all the independent variables. Get estimated coefficients. Based on your estimates, forecast one-step-ahead forecasts for all your independent variables from January 2020 on. Report all AR(1) regressions
      3) Using the estimated coefficients from 1) and the forecasts for the independent variables from 2), compute the one-step-ahead forecast St+i, for i=January 2020, End of sample.
      4) Using a simple AR(1) model for ef,t forecast St+i, for i=January 2020, End of sample.
      5) Compute the out-of-sample MSE for 3) and 4), and compare it with the RW's MSE. Test the equality of MSEs (You need to do three MGN/HLN tests: 3) vs RW; 4) vs RW; & 3) vs 4). In the MGN/HLN test make sure the model with the highest MSE is et(1).)
      Estimation Period: From the start of your sample to December 2019.
      Validation Sample: The period from January 2020 till the End of your sample.


    4. Writing up your Research Project
  • Show the plot of ef,t against (IDC,t - IFC,t).
  • Report the regression results for the whole sample. Report your results from B.2-B.8. Are you comfortable with the adequacy of the regression model?
  • Report the individual t-tests and the joint F-test. Clearly state if you reject or cannot reject PPP at the 5% level.
  • Report the in sample MSE=RSS/T for the regression model and the Random Walk models. Clearly state which model you prefer.
  • Report the selected model from B.12-B13. Clearly state why you select the variables and the results from your t-test using White/NW SE.
  • Report the regression results for your model for the Estimation period, ending in Dec 2019. Report the MSE for the validation periods (Jan 2020-on) model forecasts. Using a graph, show the forecasted St+i, along with the actual St+i, for the forecasts.
  • Report the the AR(1) regression for the Estimation period, ending in Dec 2019. Report the MSE for the validation periods (Jan 2020-on) AR(1) forecasts. Using a graph, show the forecasted St+i, along with the actual St+i, for the out-of-sample period (Jan. 2020-on).
  • Report the MSE of the RW. Compare the MSEs for your forecasts for your model, the AR(1) model and the RW. Report the 3 MGN tests. Which forecasts are significantly better?
  • Write a brief paragraph with your views about the PPP Model. (THIS PARAGRAPH IS NOT GRADED, write your opinion based on what you have done and learned.)
  • Do not e-mail me the data files (I'm not going to open them, nor give you credit for them).


    Simple Tips:
  • Do not forget that the regression uses log changes. That is, before doing the regression, you need to calculate ef, and IDC, IFC. Later you need to compute the log changes for USD Index, Oil prices, and Gold prices.
  • To calculate the Mean Square Error (MSE), you need to divide the sum of square errors by the number of errors you are using in the sum. The lower the Mean Square Error (MSE), the better the model.