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Application of the Simple Linear Regression Model: The Quantity Theory of Money 1. Statement of Economic Theory The quantity theory of money states that: “If the velocity of money is constant, then a change in the money supply must cause a proportionate change in nominal GDP. That is, the quantity of money determines the dollar value of the economy’s output.” The quantity theory of money is derived from the quantity equation: M*V = P*Y (1) M = money supply V = velocity of money – the number of times a typical dollar bill changes hands in a year. P = price level – GDP Deflator Y = real GDP – total output of economy The quantity equation is simply an identity, it is not a theory. It simply says that the number of dollars exchanged in a year, M*V, must equal the value of all goods purchased in a year, P*Y. We can turn the quantity equation into a theory by making the assumption that the velocity of money, V, is constant. Once we assume velocity is constant, the quantity equation becomes a theory of the determination on nominal income. Namely, if V is constant, then proportionate changes in the money supply must lead to proportionate changes in nominal GDP. For example if the money supply increases by 10%, the quantity theory predicts nominal GDP (P*Y) must increase by 10%. Because the inflation rate is the percentage change in the price level, this theory is also a theory of inflation. Specifically, the quantity equation 1.1 can be written in % change form as: % Change in M + % Change in V = % Change in P + % Change in Y (2) Furthermore, since we assume V is constant, our theory predicts that: % Change in M = % Change in P + % Change in Y or % Change in P = % Change in M - % Change in Y (3) In words this says that the inflation rate equals the growth rate of money minus the growth rate of real GDP. Finally, if GDP grows at a constant % each year then this can be written as: % Change in P = % Change in M - Constant. (4) Where the constant is the constant growth rate of real GDP. 1 2. Derive Testable Hypothesis The simple form of the Quantity Theory of Money leads to the following testable hypotheses: • The inflation rate should be positively related to the growth rate of the money supply. • A 1% increase in the money supply should cause a 1% increase in the inflation rate. • The constant should be negative 3. Specification of the econometric model. Our next step is to develop an empirical model capable of testing the theory. In particular ,we could write the empirical counterpart to equation (4) as: %ΔPi = β 0 + β1 %ΔM i + ε i (5) where β 0 is the intercept and equals the constant in equation (4), β 1 is the population slope parameter and ε is a random disturbance term. Equation 5 is the empirical counterpart to the theoretical model in 4. Note that the hypotheses derived in step 2 imply that β 0 should be negative (the constant in equation 4 should be negative) and that β 1 (the slope of the line) should equal 1. 4. Collection of Data To test the theory, we need to collect the relevant data. In this example, what we need is data on inflation rates and money supply growth. Or equivalently, data on prices and the money supply over time. Sheet 2 of the file money.xls contains such data for 29 countries over the period 1980-1990. Recall, that a regression, or OLS, tries to fit a line through our data. That is, OLS attempt to fit a line through the data that minimizes the sum of squared errors. To get an idea of what our data looks like, and hence how well we might expect to fit the data, I’ve graphed the inflation rate as a function of the growth rate in the money supply. Note that there appears to exist a positive relationship between the growth rate of the money supply and the inflation rate. That finding is consistent with the Quantity Theory of Money. Also note that the data appear to lie pretty much on a straight line. Thus, we would expect the data to fit relatively well with a high Rsq and a correlation coefficient close to 1. 2 Figure 1: Scatter Plot of the Relationship between the Percent Change in the Money Supply and Inflation 1000 100 Inflation Rate 10 1 1 10 100 1000 % Change in Money Supply 5. Estimation of the Model The next step is to estimate the model. Using EXCEL we could regress the inflation rate on money growth rates by going to “tools” then “data analysis” then “regression.” Using those commands, I obtained the following results: 3 Regression Statistics Multiple R 0.989768971 R Square 0.979642617 Adjusted R Square 0.97885964 Standard Error 22.23330717 Observations 28 ANOVA df SS MS Regression 1 618482.1882 618482.188 Residual 26 12852.31864 494.319948 Total 27 631334.5069 Coefficients Standard Error t Stat Intercept -3.67421468 4.563613256 -0.80511088 Money 0.981454945 0.027746668 35.371993 Note that the Intercept corresponds to b0 and Money corresponds to b1 . Thus, our “best guess” or estimate of β 0 is -3.67 and our estimate of β 1 is 0.98. These estimates are consistent with our predictions. In particular, our hypotheses were that the constant should be negative (we find that it equals -3.67) and that the slope should equal one (we find that it equals 0.98). Furthermore, consistent with Figure 1, the Rsq of the regression is quite high (equal to 0.989). Thus, 98% of the variation in inflation rates appears to be explained by variation in the money supply. Our estimate of β 1 implies that a one percent change in the money supply leads to a 0.98 percent change in the inflation rate. That estimate is very close to our hypothesis that a one percent change in the money supply leads to a one percent change in the inflation rate. We can also conduct a formal test of the hypothesis that β 1 equal one. Specifically: H 0 : β1 = 1 (1) H 1 : β1 ≠ 1 b1 − β1 0.98 − 1 (2) t= = = −.71 se(b1 ) .028 (3) Choose a 5% significant level (4) Reject H 0 if | t |> tα / 2, n − 2 4