You may recall the following well known compound

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You may recall the following well-known compound interest formula, in discrete form, from your introductory course in economics. 5 (A2.1) t 0 Y Y (1 ) t r where r is the compound (i.e., over time) rate of growth of Y . Taking the natural logarithm of (A2.1) produces: (A2.2) t 0 lnY lnY ln(1 ) t r Letting 0 0 lnY and 1 ln(1 ) r , model becomes: t 0 1 lnY t Adding the disturbance terms we get the following log-lin model, 5 The continuous form is t 0 Y Y rt e .
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ECON 301 - Introduction to Econometrics I April, 2013 METU - Department of Economics Instructor: Dr. Ozan ERUYGUR e-mail: [email protected] Lecture Notes 22 (A2.3) t 0 1 lnY t t u where the independent variable (regressor) i s “time,” which will take values of 1, 2, 3, etc. The coefficient of the trend variable in the growth model (A2.3), 1 , gives the instantaneous (at a point in time) rate of growth 6 and not the compound (over a period of time) rate of growth. But the latter can be easily found from 1 ln(1 ) r as 1 1 r e . Example To illustrate the growth model (A2.3), consider the regression model as follows: ln 7.7890 0.00743 t EXS t 2 R =0.9894 (A2.4) t-stat (3387.619) (44.2826) where EXS stands for expenditure on services, and t =1993:Q1 to 1998:Q3. The interpretation of Eq. (A2.4) is that over the quarterly period 1993:1 to 1998:3, expenditure on services increased at the (quarterly) rate of 0.743 percent (quarterly rate of growth). 6 t t t t t 1 t t Y lnY Y Y Y 1 = Y Y d d d dt dt dt .
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ECON 301 - Introduction to Econometrics I April, 2013 METU - Department of Economics Instructor: Dr. Ozan ERUYGUR e-mail: [email protected] Lecture Notes 23 For our illustrative example, the estimated slope coefficient is 0.00743. Therefore, 0.00743 1 r e = 0.00746 or 0.746 percent. Thus, in the illustrative example, the compound rate of growth on expenditure on services was about 0.746 percent per quarter, which is slightly higher than the instantaneous growth rate of 0.743 percent. This is of course due to the compounding effect.
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