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Unformatted text preview: EC3062 ECONOMETRICS
LIMITED DEPENDENT VARIABLES
Logistic Trends
One way of modelling a process of bounded growth is via a logistic
function. See Figure 1. This has been used to model the growth of a
population of animals in an environment with limited food resources. The
simplest version of the function is
(1) ex
1
=
.
π (x) =
1 + e−x
1 + ex The second expression comes from multiplying top and bottom of the ﬁrst
expression by ex .
For large negative values of x, the term 1 + ex , in the denominator of
the second expression, hardly diﬀers from 1. Therefore, when x is negative,
the logistic function resembles an exponential function.
When x = 0, there is is (1 + ex x = 0) = 2, and there is an inﬂection
as the rate of increase in π begins to decline. Thereafter, the rate of
increase declines rapidly toward zero, with the eﬀect that the value of π
never exceeds unity.
1 EC3062 ECONOMETRICS
The inverse mapping x = x(π ) is easily derived. Consider (2) ex
1 + ex
−
1−π =
1 + ex
1 + ex
1
π
=
= x.
1 + ex
e This is rearranged to give
(3) π
,
1−π ex = whence the inverse function is found by taking natural logarithms:
(4) x(π ) = ln 2 π
.
1−π EC3062 ECONOMETRICS 1.0 0.5 0.25 −4 −2 2 4 Figure 1. The logistic function ex /(1 + ex ) and its derivative. For
large negative values of x, the function and its derivative are close. In
the case of the exponential function ex , they coincide for all values of
x. 3 EC3062 ECONOMETRICS
The logistic curve needs to be elaborated before it can be ﬁtted ﬂexibly to a set of observations y1 , . . . , yn tending to an upper asymptote. The
general from of the function is
(5) γ
γeh(t)
=
;
y (t) =
−h(t)
h(t)
1+e
1+e h(t) = α + βt. Here γ is the upper asymptote of the function, and β and α determine the
rate of ascent of the function and the mid point of its ascent. It can be
seen that
(6) ln y (t)
γ − y (t) = h(t). With the inclusion of a residual term, the equation bcomes
(7) ln yt
γ − yt = α + βt + et . For a given value of γ , one may calculate the value of the dependent
variable on the LHS. Then the values of α and β may be found by leastsquares regression.
4 EC3062 ECONOMETRICS
The value of γ may also be determined according to the criterion of
minimising the sum of squares of the residuals. A crude procedure would
entail running numerous regressions, each with a diﬀerent value for γ .
The deﬁnitive value would be the one from the regression with the least
residual sum of squares.
There are other procedures for ﬁnding the minimising value of γ
of a more systematic and eﬃcient nature which might be used instead.
Amongst these are the methods of Golden Section Search and Fibonnaci
Search which are presented in many texts of numerical analysis.
The objection may be raised that the domain of the logistic function
is the entire real line—which spans all of time from creation to eternity—
whereas the sales history of a consumer durable dates only from the time
when it is introduced to the market.
The problem might be overcome by replacing the time variable t in
equation (15) by its logarithm and by allowing t to take only nonnegative
values. See Figure 2. Then, whilst t ∈ [0, ∞), we still have ln(t) ∈
(−∞, ∞), which is the entire domain of the logistic function. 5 EC3062 ECONOMETRICS 1.0
0.8
0.6
0.4
0.2 1 2 3 4 Figure 2. The function y (t) = γ/(1 + exp{α − β ln(t)}) with
γ = 1, α = 4 and β = 7. The positive values of t are the domain of
the function. 6 EC3062 ECONOMETRICS 1.0
0.8
0.6
0.4
0.2 0.5 1.0 1.5 2.0 2.5 3.0 Figure 3. The cumulative lognormal distribution. The logarithm
of the lognormal variate is a standard normal variate. 7 EC3062 ECONOMETRICS
A Binary Dependent Variable: A Probit Model in Biology
Consider the eﬀects of a pesticide on a sample of insects. For the ith
insect, the lethal dosage is the quantity δi , with log(δi ) = λi ∼ N (λ, σ 2 ).
If an insect is selected at random and is subjected to the dosage di ,
then the probability that it will die is P (λi ≤ xi ), where xi = log(di ). The
probability is
xi (8) π (xi ) = N (ζ ; λ, σ 2 )dζ. −∞ The function π (xi ) with xi = log(di ) also indicates the fraction of the
insects expected to die when all the individuals were subjected to the
same global dosage d = di .
Let yi = 1 if the ith insect dies and yi = 0 if it survives. Then the
situation of the ith insect is summarised by
(9) yi = 0, if λi > xi or, equivalently, δi > di ; 1, if λi ≤ xi or, equivalently, δi ≤ di . 8 EC3062 ECONOMETRICS
The integral of (8) may be expressed in terms of a standard normal
density function N (ε; 0, 1). Thus
P (λi < xi )
(10) with λi ∼ N (λ, σ 2 ) is equal to
P xi − λ
λi − λ
= εi < hi =
σ
σ with εi ∼ N (0, 1). Moreover, the standardised variable hi , which corresponds to the dose
received by the ith insect, can be written as
xi − λ
= β0 + β1 xi ,
σ
λ
1
and β1 = .
where β0 = −
σ
σ hi =
(11) To ﬁt the model to the data, it is necessary only to estimate the parameters
λ and σ 2 of the normal probability density function or, equivalently, to
estimate the parameters β0 and β1 .
9 EC3062 ECONOMETRICS y, 1 − π
y =1 0.3
y =0 ξi λ i* λ ξ i, λ i Figure 4. The probability of the threshold λi ∼ N (λ, σ 2 ) falling short of the
realised value λ∗ is the area of the shaded region in the lower diagram.
i 10 EC3062 ECONOMETRICS
The Probit Model in Econometrics
If the stimulus ξi exceeds the realised threshold λ∗ , then the step
i
function, indicated by the arrows in the upper diagram, delivers y = 1.
The upper diagram also shows the cumulative probability distribution
function, which indicates a probability value of P (λi < λ∗ ) = 1 − πi = 0.3
i
In econometrics, the Probit model is commonly used in describing
binary choices.
The systematic inﬂuences aﬀecting the outcome for the ith consumer
may be represented by a function ξi = ξ (x1i , . . . , xni ), which may be
a linear combination of the variables. The idiosyncratic eﬀects can be
represented by a normal random variable of zero mean.
The ith individual will have a positive response yi = 1 only if the
stimulus ξi exceeds their own threshold value λi ∼ N (λ, σ 2 ), which is
assumed to deviate at random from the level of a global threshold λ.
Otherwise, there will be no response, indicated by yi = 0. Thus
(12) yi = 0, if λi > ξi ;
1, if λi ≤ ξi . These circumstances are illustrated in Figure 4.
11 EC3062 ECONOMETRICS
The accompanying probability statements, expressed in term of a
standard normal variate, are that
(13)
λi − λ
ξi − λ
P (yi = 0ξi ) = P
= −εi >
and
σ
σ
P (yi = 1ξi ) = P ξi − λ
λi − λ
= −εi ≤
σ
σ , where εi ∼ N (0, 1). On the assumption that ξ = ξ (x1 , . . . , xn ) is a linear function, these can
be written as
(14) ∗
P (yi = 0) = P (0 > yi = β0 + xi1 β1 + · · · + xik βk + εi ) and ∗
P (yi = 1) = P (0 ≤ yi = β0 + xi1 β1 + · · · + xik βk + εi ) , where ξ (x1i , . . . , xki ) − λ
.
σ
Thus, the original statements relating to the distribution N (λi ; λ, σ 2 ) can
be converted to equivalent statements expressed in terms of the standard
normal distribution N (εi ; 0, 1).
β0 + xi1 β1 + · · · + xik βk = 12 EC3062 ECONOMETRICS
The essential quantities that require to be computed in the process
of ﬁtting the model to the data of the individual respondents, who are
indexed by i = 1, . . . , N , are the probability values
(15) P (yi = 0) = 1 − πi = Φ(β0 + xi1 β1 + · · · + xik βk ), where Φ denotes the cumulative standard normal distribution function.
These probability values depend on the coeﬃcients β0 , β1 , . . . , βk of the
linear combination of the variables inﬂuencing the response.
Estimation with Individual Data
Imagine that we have a sample of observations (yi , xi. ); i = 1, . . . , N ,
where yi ∈ {0, 1} for all i. Then, assuming that the events aﬀecting
the individuals are statistically independent and taking πi = π (xi. , β ) to
represent the probability that the event will aﬀect the ith individual, we
can write represent the likelihood function for the sample as
N (16) N
y
πi i (1 L(β ) = − πi )1−yi = i=1 i=1 13 πi
1 − πi yi (1 − πi ). EC3062 ECONOMETRICS
This is the product of n point binomials. The log of the likelihood function
is given by
N (17) yi log log L =
i=1 πi
1 − πi N log(1 − πi ). +
i=1 Diﬀerentiating log L with respect to βj , which is the j th element of the
parameter vector β , yields
∂ log L
=
∂βj
(18) N i=1
N =
i=1 N ∂πi
yi
1 ∂πi
−
πi (1 − πi ) ∂βj
1 − πi ∂βj
i=1
yi − πi ∂πi
.
πi (1 − πi ) ∂βj To obtain the secondorder derivatives which are also needed, it is
helpful to write the ﬁnal expression of (20) as
(19) ∂ log L
=
∂βj i 1 − yi
yi
−
πi
1 − πi
14 ∂ πi
.
∂βj EC3062 ECONOMETRICS
Then it can be seen more easily that
(20)
1 − yi
yi
∂ 2 πi
∂ 2 log L
=
−
−
∂βj βk
πi
1 − πi ∂βj βk
i i 1 − yi
yi
2 + (1 − π )2
πi
i ∂ πi ∂πi
.
∂βj ∂βk The negative of the expected value of the matrix of second derivatives is
the information matrix whose inverse provides the asymptotic dispersion
matrix of the maximumlikelihood estimates. The expected value of the
expression above is found by taking E (yi ) = πi . On taking expectations,
the ﬁrst term of the RHS of (20) vanishes and the second term is simpliﬁed,
with the result that
(21) E ∂ 2 log L
∂βj βk =
i 1
∂πi ∂πi
.
πi (1 − πi ) ∂βj ∂βk The maximumlikelihood estimates are the values which satisfy the
conditions
(22) ∂ log L(β )
= 0.
∂β
15 EC3062 ECONOMETRICS
To solve this equation requires an iterative procedure. The Newton–
Raphson procedure serves the purpose.
The Newton–Raphson Procedure
A common procedure for ﬁnding the solution or root of a nonlinear
equation α(x) = 0 is the Newton–Raphson procedure which depends upon
approximating the curve y = α(x) by its tangent at a point near the root.
Let this point be [x0 , α(x0 )]. Then the equation of the tangent is
(23) y = α(x0 ) + ∂α(x0 )
(x − x0 )
∂x and, on setting y = 0, we ﬁnd that this line intersects the xaxis at
(24) ∂ α(x0 )
x1 = x0 −
∂x −1 α(x0 ). If x0 is close to the root λ of the equation α(x) = 0, then we can expect
x1 to be closer still. To ﬁnd an accurate approximation to λ, we generate
16 EC3062 ECONOMETRICS
a sequence of approximations {x0 , x1 , . . . , xr , xr+1 , . . .} according to the
algorithm
(25) xr+1 ∂ α(xr )
= xr −
∂x −1 α(xr ). The Newton–Raphson procedure is readily adapted to the problem of
ﬁnding the value of the vector β which satisﬁes the equation ∂ log L(β )/∂β
= 0, which is the ﬁrstorder condition for the maximisation of the loglikelihood function. Let β consist of two elements β0 and β1 . Then the
algorithm by which the (r +1)th approximation to the solution is obtained
from the rth approximation is speciﬁed by 2 ∂ log L ∂ 2 log L −1 ∂ log L β0
β0 ∂β 2
∂β0 β1 ∂β0 0 .
= −
(26) ∂ 2 log L ∂ 2 log L ∂ log L β1 (r+1)
β1 (r)
2
∂β1
∂β1 β0
∂β1
(r )
It is common to replace the matrix of secondorder partial derivatives
in this algorithm by its expected value which is the negative of information
17 EC3062 ECONOMETRICS y x1
x
x0 x2 Figure 5. If x0 is close to the root of the equation
α(x) = 0, then we can expect x1 to be closer still. matrix. The modiﬁed procedure is known as Fisher’s method of scoring.
18 EC3062 ECONOMETRICS
The algebra is often simpliﬁed by replacing the derivatives by their expectations, whereas the properties of the algorithm are hardly aﬀected.
In the case of the simple probit model, where there is no closedform
expression for the likelihood function, the probability values, together with
the various derivatives and expected derivatives to be found under (18)
to (21), which are needed in order to implement one or other of these
estimation procedures, may be evaluated with the help of tables which
can be read into the computer.
Recall that the probability values π are speciﬁed by the cumulative
normal distribution
h (27) π (h) =
−∞ 2
1
√ e−ζ /2 dζ.
2π We may assume, for the sake of a simple illustration, that the function
h(x) is linear:
(28) h(x) = β0 + β1 x.
19 EC3062 ECONOMETRICS
Then the derivatives ∂πi /∂βj become
(29)
∂πi ∂h
∂πi
∂πi ∂h
∂πi
=
= N {h(xi )} and
=
= N {h(xi )}xi ,
.
.
∂β0
∂h ∂β0
∂β1
∂h ∂β1
where N denotes the normal density function which is the derivative of π .
Estimation with Grouped Data
In the classical applications of probit analysis, the data was usually in
the form of grouped observations. Thus, to assess the eﬀectiveness of an
insecticide, various levels of dosage dj ; j = 1, . . . , J would be administered
to batches of nj insects. The numbers mj = i yij killed in each batch
would be recorded and their proportions pj = mj /nj would be calculated.
If a suﬃciently wide range of dosages are investigated, and if the
numbers nj in the groups are large enough to allow the sample proportions
pj accurately to reﬂect the underlying probabilities πj , then the plot of
pj against xj = log dj should give a clear impression of the underlying
distribution function π = π {h(x)}.
In the case of a single experimental variable x, it would be a simple
matter to infer the parameters of the function h = β0 + β1 x from the plot.
20 EC3062 ECONOMETRICS
According to the model, we have
(30) π (h) = π (β0 + β1 x). From the inverse h = π −1 (π ) of the function π = π (h), one may obtain the
values hj = π −1 (pj ). In the case of the probit model, this is a matter of
referring to the table of the standard normal distribution. The values of π
or p are found in the body of the table whilst the corresponding values of
h are the entries in the margin. Given the points (hj , xj ) for j = 1, . . . J ,
it is a simple matter to ﬁt a regression equation in the form of
(31) hj = b0 + b1 xj + ej . In the early days of probit analysis, before the advent of the electronic
computer, such ﬁtting was often performed by eye with the help of a
ruler.
To derive a more sophisticated and eﬃcient method of estimating the
parameters of the model, we may pursue a method of maximumlikelihood.
This method is a straightforward generalisation of the one which we have
applied to individual data.
21 EC3062 ECONOMETRICS
Consider a group of n individuals which are subject to the same probability P (y = 1) = π for the event in question. The probability that the
event will occur in m out of n cases is given by the binomial formula:
(32) B (m, n, π ) = n!
nm
n−m
π m (1 − π )n−m .
=
π (1 − π )
m!(n − m)!
m If there are J independent groups, then the joint probability of their outcomes m1 , . . . , mj is the product
(33)
J L=
j =1 J nj
nj
m
πj j (1 − πj )nj −mj =
mj
mj
j =1 πj
1 − πj mj (1 − πj )nj . Therefore the log of the likelihood function is
J (34) mj log log L =
j =1 πj
1 − πj + nj log(1 − πj ) + log nj
mj . Given that πj = π (xj. , β ), the problem is to estimate β by ﬁnding the
value which satisﬁes the ﬁrstorder condition for maximising the likelihood
22 EC3062 ECONOMETRICS
function which is
∂ log L(β )
= 0.
∂β (35) To provide a simple example, let us take the linear logistic model
eβ0 +β1 x
.
π=
1 + eβ0 +β1 x (36) The socalled logodds ratio is
(37) log π
1−π = β0 + β1 x. Therefore the loglikelihood function of (34) becomes
J mj (β0 + β1 xj ) − nj log(1 − eβ0 +β1 xj ) + log (38) log L =
j =1 23 nj
mj , EC3062 ECONOMETRICS
and its derivatives in respect of β0 and β1 are
(39)
eβ0 +β1 xj
∂ log L
mj − n j
=
=
∂β0
1 + eβ0 +β1 xj
j
∂ log L
=
∂β1 mj xj − nj xj
j eβ0 +β1 xj
1 + eβ0 +β1 xj (mj − nj πj ),
j xj (mj − nj πj ). =
j The information matrix, which, together with the above derivatives, is
used in estimating the parameters by Fisher’s method of scoring, is provided by
j (40)
j mj πj (1 − πj ) j mj xj πj (1 − πj ) mj xj πj (1 − πj ) mj x2 πj (1
j
j 24 − πj ) . ...
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 Spring '12
 D.S.G.Pollock
 Econometrics

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