Assuming log utility, the basic consumption Euler equation can be written

C_t+1 /C_t =(1+r_t)

If we take logs of this, and use the approximation that the natural log of one plus a small number is approximately the small number, then we have:

r_t= g^c_t+1 - ln

In other words, the real interest rate ought to equal the expected growth rate of consumption minus the log of the discount factor.

(a)For the period 1947 through 2018, download annual data on the GDP price deflator (https://fred.stlouisfed.org/series/A191RI1A225NBEA),

annual data on real consumption growth (https://fred.stlouisfed.org/series/DPCERL1A225NBEA),

and data on the 3-Month Treasury Bill rate (https://fred.stlouisfed.org/series/TB3MS),

this series is available at a higher frequency than annual, so to get it in annual terms, click "edit graph" and modify frequency to annual using the average method). The approximate real interest rate is r_t = i_t -^e _t+1

Measure i_t by the 3-Month T-Bill rate and assume expected inflation equals realized inflation one period ahead (i.e. the interest rate observation in 1947 will be the 3-Month T-Bill in 1947, while you will use realized inflation in 1948 for expected inflation in 1947). Compute a series for the real interest rate. Plot this series. What is the average real interest rate? How often has it been negative? Has it been negative or positive in the last 10 years?

What is the correlation between the real interest rate series you create and expected consumption growth (i.e. compute the correlation between consumption growth in 1948:2018 and the real interest rate between 1947:2017)? Is the sign of this correlation qualitatively in line with the predictions of the Euler equation? Is this correlation strong?