The Textbook Shows That Ln(x + ΔX)- Ln(x)≅ $\frac { \Delta x } { x }$

question 59

Essay

The textbook shows that ln(x + Δx)- ln(x)≅ $\frac { \Delta x } { x }$ Show that this is equivalent to the following approximation ln(1 + y)≅ y if y is small. You use this idea to estimate a demand for money function, which is of the form m = β₀ × $G D P ^ { \beta 1 }$ ×, $( 1 + R ) ^ { \beta _ { 1 } }$ × e^u where m is the quantity of (real)money, GDP 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 GDP, and on (1+R). Can you for see a problem in using this transformation?

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Definitions:

Creatine Phosphate

A high-energy reserve molecule in muscles, used to rapidly generate ATP during the first few seconds of intense physical activity.

Lactate Concentration

The level of lactate, a metabolic byproduct of anaerobic respiration, present in the blood or muscles, often measured during physical exertion.

Tension

A physical force applied to something that tends to stretch it, often leading to stress or strain on the material.

Mitochondria

Organelles within cells that generate most of the cell's supply of adenosine triphosphate (ATP), used as a source of chemical energy, and involved in other processes such as signaling and cellular differentiation.