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Application of the Simple Linear Regression Model The Quantity by kellena99


									                          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


        % 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.

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.

Figure 1:      Scatter Plot of the Relationship between the Percent Change in the Money Supply
               and Inflation


                Inflation Rate


                                        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:

                          Regression Statistics
                Multiple R                 0.989768971
                R Square                   0.979642617
                Adjusted R Square            0.97885964
                Standard Error             22.23330717
                Observations                         28

                                                 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
        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


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