CDS M Phil Econometrics

Transcription

1 CDS M Phil Econometrics an Pillai N 21/10/ CDS M Phil Econometrics Functional Forms and Growth Rates 2 CDS M Phil Econometrics 1

2 Beta Coefficients standardized coefficient or beta coefficient Idea is to replace y and each x variable with a standardized version i.e. subtract mean and divide by standard deviation Coefficient reflects standard deviation of y for a one standard deviation change in x Functional Form OLS can be used for relationships that are not strictly linear in x and y by using nonlinear functions of x and y will still be linear in the parameters the natural log of x, y or both quadratic forms of x interactions of x variables 2

3 Interpretation of Log Models Double-log model: If the model is ln(y) = b 0 + b 1 ln(x) + u b 1 is the elasticity of y with respect to x Interpretation of Log Models Semi-log models: (1) If the model is ln(y) = b 0 + b 1 x + u b 1 is approximately the percentage change in y given a 1 unit change in x If x = time, b 1 = growth rate 3

4 Interpretation of Log Models Semi-log models: (2) If the model is y = b 0 + b 1 ln(x) + u b 1 is approximately the change in y for a 100 percent change in x Used in Engel expenditure models: German statistician, Ernst Engel ( ): Total expenditure on food tends to increase in AP as total expenditure increases in GP Why use log models? They give a direct estimate of elasticity For models with y > 0, the conditional distribution is often heteroskedastic or skewed, while ln(y) is much less so The distribution of ln(y) is more narrow, limiting the effect of outliers 4

5 Some Rules of Thumb What types of variables are often used in log form? Dollar amounts that must be positive Very large variables, such as population What types of variables are often used in level form? Variables measured in years Variables that are a proportion or percent y i Reciprocal model 1 x i u i dy dx Slope everywhere negative and decreases in absolute value as x increases. As x 0, y, and as x, y. 2 x y > 0 > 0 x y - > 0 < 0 x y / < 0 > 0 x 5

6 Reciprocal model: Phillips Curve Percentage rate of change of money wages (y) and unemployment rate of the UK for AW Phillips (1958) Economica Vol 15: Rate of change of money wages % 0 - The natural rate of unemployment Unemployment rate % Logarithmic reciprocal model lny i 1 x dy dx i u y 2 x i y i e Slope is positive for positive x ui xi 2 d y y 2 dx x x Point of inflexion at x = /2. e /2 6

7 Logarithmic reciprocal model Point of inflexion at x = /2. y e To the left of this point, slope with x; to the right, slope. y i e ui xi As x, y e /2 x Quadratic Models y = x + 2 x 2 + u What s the slope here? Not 1 alone dy dx x 7

8 More on Quadratic Models For 1 0 and the turning point is at 2 0, x * 1 2 2, which is the same as when 1 0 and 2 0 Interaction Terms y = b 0 + b 1 x 1 + b 2 x 2 + b 3 x 1 x 2 + u What s the slope wrt x 1? Not b 1 alone dy dx x 2, so to summarize the effect of we typically evaluate the above at x 1 on y x 2 8

9 Simple Compound Exponential Kinked exponential Growth rates Let us start with an application of Difference equations 17 Compound Interest: Applications of DE Amount deposited last year: A t 1. Rate of interest: r Amount obtainable this year: A t = (1+r) A t 1, or A t (1+r) A t 1 = 0, a DE(1). What is its time path? The compound interest formula: A t = A 0 (1+r) t What is the nature of the time path? 18 9

10 The compound interest formula: A t = A 0 (1+r) t : An initial amount A 0 grows into a final amount A t in t years. What is the growth rate? r : Average annual compound growth rate! A t is compound growth function. 19 Consider the compound interest DE(1): Rewriting, A t A t 1 = ra t 1 ; or A t = ra t 1. A t (1+r) A t 1 = 0. Change in A is proportional to its previous value, r being the constant of proportionality. Then A t /A t 1 = (A t A t 1 )/A t 1 = r: the growth rate! 20 10

11 The compound growth function: A t = A 0 (1+r) t Given A t, A 0 and t, How to obtain r, average annual compound growth rate? r = (A t /A 0 ) (1/t) 1; or ln( At / A0 ) r exp 1 t 21 Simple (annual) growth rate: Annual rate of change; given two consecutive years, t = 1 and 0: Change: A = A 1 A 0 Rate of change: A/A 0 = (A 1 A 0 )/A 0 A 1 = 1 A0 Given t years, average annual growth rate = At 1 ( At A0 )/ A0 A0 t t 22 11

12 Year Index Annual growth rate For 2001: r = ( )/100 = 0.1 or 10%; For 2002: r = ( )/110 = or For 2003: 13.6%; r = ( )/125 = 0.2 or 20%; For 2004: r = ( )/150 = 0.2 or 20% 23 Year Index Annual growth rate Average annual growth rates Simple and Compound: No. of years, t = = Simple r = ( )/100 4 or 20% Compound r = = = or 15.8% 24 12

13 Note that the growth rate in both the cases is based on two points in time. It does not take into account the intermediate values of the series. 25 Note the difference: Which one to take? Simple growth rate: 20%; Compound growth rate: 15.8%. Economic variables in their growth almost invariably obey the law of compound interest, not simple interest. Why? Take for example, the growth of an index (inflation, production, etc.) 26 13

14 Growth of a Price Index Say, inflation is 10% a year Base year (0) index: Year 1 index: 110 (10% over ) 0 I n d e x Year 2 index: 121 (10% over 110). A compound growth! Index Growth is compounded because each year s 10% rise is applied to a level that is increasing every year. 27 Year Discrete time vs. continuous time So far time taken as discrete: Compound growth: A t = A 0 (1+r) t With continuous time: Exponential growth or Geometric growth: A(t) = A(0)e rt where e = is the base of natural exponential function

15 Continuous time: Exponential Growth As in the discrete time case A t A t 1 = ra t 1 ; or A t = ra t 1. Change in A is proportional to its value, r being the constant of proportionality. Then the rate of change da/dt = ra. Rewriting, we get r as the growth rate: 1 A da dt 29 r Note da/dt = ra can be rewritten as da/dt ra = 0, a homogeneous first order differential equation. Its solution is given by A(t) = A(0)e rt, the exponential growth function

16 The general principle behind exponential growth: the larger a number gets, the faster it grows. Any exponentially growing number will eventually grow larger than any other number which grows at only a constant rate for the same amount of time. This is demonstrated by a classic riddle: 31 The classic riddle: A child is offered two choices for an increasing weekly allowance: the first option begins at Re.1 and increases by Re. 1 each week, while the second option begins at 1 paise and doubles each week

17 Option 1 : Option 2 : Week Rs Ps Option 1 : Week Rs eventually grows much larger. The first option, growing at a constant rate of Re. 1/week, pays more in the short run. Option 2 : Ps the second option 33 Mathematically, In the first case, the payout in week t is simply t + 1 Rupees: a linear growth. The payout grows at a constant rate of Re. 1 per week; in week 15, the payout is = Rs

18 Mathematically, In the second case, the allowance in week t is 2 t paise; thus, in week 15 the payout is 2 15 = Paise = Rs All formulas of the form k t, where k is an unchanging number greater than 1 (e.g., 2), and t is the amount of time elapsed, grow exponentially. 35 So k t, where k is constant and t is variable is an exponential function. Now consider the compound growth function: A t = A 0 (1+r) t ; The part (1+r) t can be written as k t, where k = (1+r): the compound growth function itself is exponential; The two functions: two sides of the same coin one discrete; one continuous

19 The graph illustrates how an exponential growth surpasses both linear and cubic growths: Red line: Linear growth (50t): 500, at t = 10 Blue line: Polynomial (t 3 ): 1000, at t = 10 Green line: Exponential (2 t ): 1024, at t = An Exponential story: Rice on a Chessboard A courtier presented the Persian king with a beautiful, hand-made chessboard. The king asked what he would like in return for his gift. The courtier surprised the king by asking for one grain of rice on the first square, two grains on the second, four grains on the third, etc. The king readily agreed and asked for the rice to be brought

20 An Exponential story: Rice on a Chessboard (Continued ) All went well at first, but the requirement for 2 n 1 grains on the nth square demanded over a million (1,048,576) grains on the 21st square, more than a trillion ( E+12) on the 41 st, and E+18 on the 64 th. and there simply was not enough rice in the whole world for the final squares. 39 Limitations Even when exponential growth seems slow in the short run, it becomes impressively fast in the long run, with the initial quantity doubling at the doubling time, then doubling again and again

21 Limitations (Continued.) For instance, a population growth rate of 2% per year may seem small, but it actually implies doubling after 35 years, doubling again after another 35 years (i.e. becoming 4 times the initial population), etc. 41 Doubling time The doubling time (or generation time) is the period of time required for a quantity to double in size or value. When the relative growth rate is constant, the quantity undergoes exponential growth and has a constant doubling time which can be calculated directly from the growth rate

22 Doubling time (Continued.) Consider the compound growth function: A t = A 0 (1+r) t A t becoming double A 0 A t /A 0 = 2; that is, 2 = (1+r) t ; Taking (natural) log and rewriting, t = ln(2)/ln(1 + r) = /ln(1 + r). 43 Doubling time (Continued.) Consider the exponential growth function: A t = A 0 e rt A t becoming double A 0 A t /A 0 = 2; that is, 2 = e rt ; Taking natural log and rewriting, t = ln(2)/r = /r. Note: If r is small, ln(1 + r) r

23 Doubling time (Continued.) Annual growth rate of population in India in 2001: Male: 1.71%; Female: 1.92%; Total: 1.81% Doubling time for male population: = /ln(1.0171) = / = years. Doubling time for female population: = /ln(1.0192) = years. Doubling time for total population: = /ln(1.0181) = years. 45 Growth rate (%) Doubling time (years) Growth rate (%) Doubling time (years)

24 Limitations (Continued.) So some people challenge the exponential growth model on the ground that it is valid for the short term only, i.e. nothing can grow indefinitely. 47 Exponential Growth Rate: Given Exponential growth: A(t) = A(0)e rt Then A(t)/A(0) = e rt, or rt = ln[a(t)/a(0)], or r ln[ A( t)/ t A(0)] 48 24

25 Exponential Growth rate: r ln[ A( t)/ A(0)] t Compare with compound growth rate: r ln( At / A0 ) exp 1 t Compound growth rate = Exp(exponential growth rate) Note that the growth rate in both the cases is based on two points in time only. It does not take into account the intermediate values of the series. Hence Regression (Least-squares) squares) growth rate

26 Least-squaressquares growth rate. Least-squares growth rates are used wherever there is a sufficiently long time series to permit a reliable calculation. The least-squares growth rate, r, is estimated by fitting a linear regression trend line to the logarithmic annual values of the variable in the relevant period. The regression equation takes the form ln y t = a + bt. 51 The regression equation, ln y t = a + bt may be taken as (i) the logarithmic transformation of the exponential growth equation, y(t) = y(0)e rt, with a = ln y(0) and b = r, the parameters to be estimated. If b* is the least-squares estimate of b, the same gives the average annual exponential growth rate, r, and is multiplied by 100 for expression as a percentage

27 The regression equation, ln y t = a + bt may also be taken as (ii) the logarithmic transformation of the compound growth equation, y t = y o (1 + r) t, with a = ln y o and b = ln(1 + r). If b* is the least-squares estimate of b, the average annual growth rate, r, is obtained as [exp(b*) 1]. Remember Compound growth rate = Exp(exponential growth rate) Estimating Growth rate of real state domestic product (SDP) of Kerala for : 27

28 Growth rate of real state domestic product (SDP) of Kerala for : Growth rate : Coefficient of t = = 3.52% Growth rate of real state domestic product (SDP) of Kerala for : LNSDP Kerala ln SDP = time (476) (29.4) R 2 = F(1, 38) = 862.4; n = e6 2.5e6 2e6 1.5e6 1e6 NSDP Kerala Exponential growth rate: or 3.52% per year. Compound growth rate: exp(0.0352) 1 = = or 3.58% per year. 28

29 3e6 NSDP Kerala Note the graph: 2.5e6 2e6 A break, kink, around e6 Post : higher growth rate. Two sub-period growth rates possible by estimating them separately, or, 1e t = k 57 3e6 NSDP Kerala Note the graph: 2.5e6 2e6 A break, kink, around e6 1e equivalently, by fitting the single equation: ln Y t = a 1 D 1 + a 2 D 2 + b 1 D 1 t + b 2 D 2 t + u t ; where D j = 1 in sub-period j; = 0, otherwise. t = k 58 29

30 t = k Estimating Discontinuous sub-period Growth rates of real state domestic product (SDP) of Kerala for : Discontinuous sub-period growth rates for Kerala: with kink in : 30

31 Discontinuous sub-period growth rates for Kerala: with kink in : NSDP Kerala 3e6 2.5e6 2e6 1.5e6 1e ln SDP t = D D D 1 t (651) (94.4) (22.7) D 2 t (16) R2 = ; F(3,36) = 825.4; n = The two trend lines: discontinuous. This discontinuity can be eliminated via a linear restriction: a 1 +b 1 k = a 2 + b 2 k t = k At the kink point, t = k, the two trend line values are equal: a 1 +b 1 k = a 2 + b 2 k, or a 2 = a 1 +b 1 k b 2 k. This gives the Kinked Exponential Model for growth rate estimation

32 Kinked Exponential Model for growth rate estimation Substituting for a 2 in ln Y t = a 1 D 1 + a 2 D 2 + b 1 D 1 t + b 2 D 2 t + u t, We get the restricted form: ln Y t = a 1 + b 1 (D 1 t + D 2 k) + b 2 (D 2 t D 2 k) + u t ; Note: a 1 D 1 + a 1 D 2 = a Kinked Exponential Model for growth rate estimation ln Y t = a 1 + b 1 (D 1 t + D 2 k) + b 2 (D 2 t D 2 k) + u t ; The OLS estimates of b 1 and b 2 give the exponential growth rates for the two subperiods. There is a kink between the two trend lines, whenever b 1 estimate b 2 estimate

33 a 1 +b 1 k = a 2 + b 2 k Ref: James K Boyce t = k Kinked Exponential Models for Growth Rate Estimation Oxford Bulletin of Economics and Statistics, 1986, vol. 48, issue 4, pages Estimating Kinked Exponential Growth rates of real state domestic product (SDP) of Kerala for : 33

34 Kinked Exponential growth rates for Kerala: with kink in : Test whether there is a kink between the two trend lines Kinked Exponential growth rates for Kerala: with kink in : NSDP Kerala 3e6 2.5e6 2e6 1.5e6 1e ln SDP t = (D 1 t + D 2 k) (675) (25.3) (D 2 t D 2 k) (19.9) R 2 = ; F(2,37) = 1194; n =