The Textbook Shows That $\ln ( x + \Delta x ) - \ln ( x ) \cong \frac { \Delta x } { x }$

Essay

The textbook shows that $\ln ( x + \Delta x ) - \ln ( x ) \cong \frac { \Delta x } { x }$ Show that this is equivalent to the following approximation $\ln ( 1 + y ) \cong y$ if y is small. You use this idea to estimate a demand for money function, which is of the form $m = \beta _ { 0 } \times G D P ^ { \beta _ { 1 } } \times ( 1 + R ) ^ { \beta _ { 2 } } \times e ^ { u } ,$ where m is the quantity of (real) money, G D P is the value of (real) Gross Domestic Product, and R is the nominal interest rate. You collect the quarterly data from the Federal Reserve Bank of St. Louis data bank ("FRED"), which lists the money supply and GDP in billions of dollars, prices as an index, and nominal interest rates in percentage points per year You generate the variables in your regression program as follows: m= (money supply)/price index; GDP = (Gross Domestic Product/Price Index), and R= nominal interest rate in percentage points per annum. Next you perform the log-transformations on the real money supply, real G D P , and on (1+R) . Can you for see a problem in using this transformation?