A number of equations are provided for data simulation.

In addition, there are various time-series data available, such as:

• datasets - a package of dataset collection;
• github, which are available in R’s fma package.

# Data: Historical

## Short-rate and Long-rate Models

Market practitioners general classify interest rates into short-term and long-term categories:

• Short-term - includes rates on Treasury bills, interbank trading of deposits, certificates of deposit, etc. These rates have a life span of less than one year;
• Long-term - includes rates on long-term bonds, which may have a maturity of up to 30 years.

Assume that:

- Y = Longrate = long-term interest rates;
- X = Shortrate = short-term interest rates
suppressPackageStartupMessages({require(readxl)})
txt1 <- "http://uosis.mif.vu.lt/~rlapinskas/(data%20R&GRETL/"
txt2 <- "interestrates.xls"
tmp = tempfile(fileext = ".xls")
destfile = tmp, mode = "wb")
#Read it as an excel file
interestrates <- ts(read_excel(path = tmp, col_names = TRUE)[, -1], start = 1954, frequency = 4)
• Examine whether the two classes of interest rates are in some way tied together.

## Spot and forward rate

The forward rate and spot rate are different prices, or quotes, for different contracts:

• spot rate - is a contracted price for a transaction that is taking place immediately (it is the price on the spot);
• forward rate - is the settlement price of a transaction that will not take place until a predetermined date in the future; it is a forward-looking price.
suppressPackageStartupMessages({require(readxl)})
txt1 <- "http://uosis.mif.vu.lt/~rlapinskas/(data%20R&GRETL/"
txt2 <- "forexN.xls"
tmp = tempfile(fileext = ".xls")
destfile = tmp, mode = "wb")
#Read it as an excel file
forexN <- ts(read_excel(path = tmp, col_names = TRUE), start = 1, frequency = 1)
colnames(forexN) <- c("spot", "forward")
• Examine whether forward rates are typically calculated based on the spot rate.

## German M1 money demand

Contains the following variables:

- logm1 - logarithm of real M1 per capita;
- logprice - logarithm of a price index;
- loggnp - logarithm of real per capita gross national product;
- interest - long-run interest rate.
data(M1Germany, package = "dynlm")
• Investigate the linearity and stability of German M1 money demand, logm1. For the exogenous variables, consider the remaining variables, like logprice, loggnp, interest;
• Examine how logprice depends on interest and logm1.

## US Macroeconomic Time series data

Time series data on 12 US macroeconomic variables for 1950-2000.

- gdp - Real gross domestic product (in billion USD);
- consumption - Real consumption expenditures;
- invest - Real investment by private sector;
- government - Real government expenditures;
- dpi - Real disposable personal income;
- cpi - Consumer price index;
- m1 - Nominal money stock;
- tbill - Quarterly average of month end 90 day treasury bill rate;
- unemp - Unemployment rate;
- population - Population (in million), interpolation of year end figures using constant growth rate per quarter;
- inflation - Inflation rate;
- interest - Ex post real interest rate (essentially, tbill - inflation).
data(USMacroG, package = "AER")
• Consider a Distributed lag model: how consumption responds to changes in income, dpi;
• Consider an Autoregressive distributed lag: assume that the effects of changes in income (dpi) on consumption persist through time.
• Examine how consumption depends on the gdp. If needed, take logarithms of the data.

Note: It may very well be the case that some (or even all) of the data do not have unit roots. The idea is to carry out the unit root testing and model building procedures, as you would when working with any other empirical data.

1. Plot the series - do they appear stationary. Do they appear to exhibit exponential changes? If needed, transform the series to remove any exponential growth. Continue working with the transformed data.

2. Plot their $$\rm ACF$$ and $$\rm PACF$$ - doe the series appear to be autocorrelated?

3. Examine the CCF’s - do the variables appear to be correlated with one another?

4. Carry out a unit root test on all of the appropriate variables. What do the results indicate?

5. If needed, carry out the Engle-Granger test for cointegration.

6. Depending on the previous results, estimate an appropriate model:

• A Distributed Lag Model;
• An Autoregressive Distributed Lag Model;
• An Error Correction Model;
7. Assume that one of your $$X$$ variables exhibits a temporary unit increase for period $$t$$. Then:

• What is the immediate effect on $$Y$$ (i.e. the short-run multiplier)?
• Describe the $$s$$-period delay multiplier (i.e. the effect at period $$t+s$$) - as $$s$$ increases, how does the value of $$Y$$ change?
• What is the long-run multiplier (i.e. the long-run effect)?
8. Assume that one of your $$X$$ variables exhibits a permanent unit increase after period $$t$$, so that $$... = X_{t-1} = X$$ and $$X_t = X_{t+1} = ... = X+1$$. Then:

• What is the immediate effect on $$Y$$ (i.e. the short-run multiplier)?
• Describe the $$s$$-period delay multiplier (i.e. the effect at period $$t+s$$) - as $$s$$ increases, how does the value of $$Y$$ change?
• What is the long-run multiplier (i.e. the long-run effect)?