Jamaican Coffee Supply

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Jamaican Coffee Supply

• Yu Xiangmiao Don

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• R.l. Williams, in a 1972 Social and Economic
  Studies article entitled Jamaican Coffee Supply,
  1953-1968 An Exploratory Study, studies the
  response of Jamaican coffee producers to various
  economic incentives. In general, the study
  concluded that the responses are those predicted
  by economic theory.

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Our model the Production Function

• Assume
• There are only 2 factors which can affect the
  coffee supply, which are Labour(L) and
  Capital(K).
• Therefore
• Our desired production of coffee Q is
• QQ(MPPK, MPPL)
• MMPmarginal physical product

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• Also
• MMPKc/P
• MMPLW/P
• Consequently
• Q(c/P, W/P)
  (1)
• Since we cannot measure the cost of capital, so
  we use the desired number of coffee trees
  instead.
• Q(T, W/P)
  (2)
• Where T is a function of the desired number of
  coffee trees.

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• In order to calculate T, we find that T depends
  upon the use of land for growing substitute and
  complementary crops. So T is determined by the
  price of the substitute and complementary crops.
• We assume that the complementary crop is cocoa,
  and the alternative crop is bananas.
• Therefore
• TT(PB, PC)
  (3)
• Where PB is the price of bananas,
• and PC is the price of cocoa

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• Substituting equation (3) into equation (2) gives
  one equation (4) as a general statement of
  desired output of coffee.
• QQ(P, PB, PC, W)
  (4)
• Although equation (4) represents the desired
  supply of coffee, Williams realized there would
  be lags in the expansion of coffee production to
  desired levels. Therefore, he specified a partial
  adjustment model for the actual supply of coffee.

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• Qt/Qt-1 (Qt/Qt-1)q 0ltqlt1
  (5)
• Where q represents the speed of adjustment
  coefficient.
• We take the logarithms of both sides of equation
  (5), we get
• log(Qt) qlog(Qt)(1-q)log(Qt-1) (6)
• Now, we completed the specification of our model
  by making the log of desired output a linear
  function of the logs of the price of coffee,
  price of bananas, price of cocoa, and the wage
  rate. Substitute them into equation (6) and
  adding an error term, then we get our final model.

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Our final model

• log(Qt) qaqb1log(Pt)qb2log(Pbt)
• qb3log(Pct)qb4log(Wt)
• (1-q)log(Qt-1) et
  (7)
• Where the bs coefficients in the linear
  function for the log of desired output and et a
  random error term. b1 and b3 are expected to be
  positive and b2 and b4 are expected to be
  negative.

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Problem 1

• (a)Estimate equation (7) from the text using
• the data provided. Note that Williams
• ended up lagging all the price and wage
  variables. This has already been done for you in
  the data set.

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• SHAZAM Command
• SAMPLE 2 16
• READ(D/123.TXT) S PLAG PCLAG PBLAG WLAG
• GENR lnSLOG(S)
• GENR lnPLOG(PLAG)
• GENR lnPCLOG(PCLAG)
• GENR lnPBLOG(PBLAG)
• GENR lnWLOG(WLAG)
• GENR lnQLAG(lnS)
• OLS lnS lnP lnPC lnPB lnW lnQ/loglog

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SHAZAM output
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• (b) Test your model for the existence of a
• regression at the 10 level of significance.
• We can use anf command to compute the F-value
  to see the significance.
• The 10 level of significance for this
  condition
• is F2.61gt0.9306948
• So we dont reject Ho, and our model is
  insignificance under a 10 level.

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• (c) Does it appear that you have a regression?
• From the answer to (b) part, we can say
• that we should not reject our hypothesis
• and have got a regression.

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Problem 2

• Test the model estimated in problem 1 for the
  existence of first order autocorrelation using
  the Durbin h-test.

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SHAZAM Command

• SAMPLE 2 16
• READ(D/123.TXT) S PLAG PCLAG PBLAG WLAG
• GENR lnSLOG(S)
• GENR lnPLOG(PLAG)
• GENR lnPCLOG(PCLAG)
• GENR lnPBLOG(PBLAG)
• GENR lnWLOG(WLAG)
• GENR lnQLAG(lnS)
• OLS lnS lnP lnPC lnPB lnW lnS(1.1)/DLAG

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SHAZAM OUTPUT
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• Suppose our test is at the 10 level of
• significance. It is because that P-VALUE of
• the lag variable LNQ(lnS(1.1)) is 0.105,
• which is higher than the level of significant
• we supposed, so we should reject the null
• hypothesis, that means there is the first order
• autocorrelation in our problem.

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• (b) Why didn't you use the classic Durbin- Watson
  test for autocorrelation?
• It is because that Durbin-Watson d statistics
  is not suit for estimating the existence of the
  first order autocorrelation. It is because that
  in this kind of models, the d statistical value
  is always bias to 2, since 2 is the expect value
  of d value. In another word, if we use the
  classical method to calculate d statistical
  value, it would be biased for us to detect the
  autocorrelation in the models. So we choose h
  statistics for the large-sample instead.

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• (c) Were you able to compute the Durbin h-
• statistic? Why or why not?
• From the output, we can see the standard error
• of LNQ(lnS(1.1)) is 0.4499, so
• n0.449950,44992.2495gt1,
• also h(1-0.5d)(n/(1-nvar(C-LNQ))1/2
• (C-LNQthe coefficient of LNQ)
• Consequently, we CANNOT calculate the
• DURBIN H STATISTIC.

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Problem 3

• (a) Test the model estimated in Problem 1 for
  first order autocorrelation using the Lagrange
  Multiplier test at the 10 level of significance.

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• First we should calculate the Fitted value
  fvln(s)
• Then we regression the function
• etyb4et-1u

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• The SHAZAM output

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• From the SHAZAM output, we can get the R2,
  which is 0.9965.
• LM Statistics(LM(c2)(n-p)R2(15-1)0.996513.
  951
• The Chi-squared is 2.70554 at the 10 level,
  while our chi-squared value is 13.951, which is
  higher than 2.70554.
• Therefore, there is evidence of
  autocorrelation.

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• (b) Does it appear that your model is plagued
  with autocorrelation?
• From Part (a), we can see there is evidence of
  autocorrelation.

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Problem 4

• (a) Reestimate equation (7) correcting it for
  first order autocorrelation using the
  Cochrane-Orcutt procedure.

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SHAZAM Command

• SAMPLE 2 16
• READ(D/123.TXT) S PLAG PCLAG PBLAG WLAG
• GENR lnSLOG(S)
• GENR lnPLOG(PLAG)
• GENR lnPCLOG(PCLAG)
• GENR lnPBLOG(PBLAG)
• GENR lnWLOG(WLAG)
• GENR lnQLAG(lnS)
• AUTO lnS lnP lnPC lnPB lnW lnS(1.1)/DLAG RSTAT

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SHAZAM OUTPUT
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• From the output, we can see that our R-square
  is improved after the adjustment of
  autocorrelation(R-square is 0.3408 in Problem 1),
  so does the t-ratio of LNP and LNPC.

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Problem 6

• What do we mean when we say a model is
  misspecified?
• A model is misspecified would mean that
• our model is totally wrong or we have ignored
• some important variables in our model.

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• (b) In our example, q being greater than 1
  implies that output growths more than desired
  during expansion periods and that output
  contracts more than desired during the desired
  contraction periods. In a simple sense, we may
  view the observed supply function as being more
  elastic than the desired supply function. From
  your knowledge of the theory of the supply
  function, what variable omitted from our
  estimated model could be causing this more
  elastic response?

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• I think the problem is the latter one. That is
• because that when we establish the model we
• dont know the total effect that our variables
• have on the output of coffee. In another word,
• we have assumed the following equation
• Qt/Qt-1 (Qt/Qt-1)q 0 lt q lt 1
• But actually, we dont know the relationship
• between Qt and Qt.

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• Suppose that the output of coffee increases
• this year than last year. So Qt/Qt-1 is higher
• than 1. Then what about Qt? Qt is just the
• output this year if there is only have 4
• effective factors. (P, PB, PC and W) So what
• is the total effect of these four factors to the
• output of coffee? In other words, what is the
• total effect of the whole of other factors
• except these four to the output of coffee?

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• Actually we dont know. Since we only
• have 15 samples, which is also too small, so it
• is quite easily to ignore this effect or appear
• some other wrong relationships we see from
• our regression.
• In our model, we assume that 0 lt q lt 1
• which means Q is lower than Qt.
• Alternatively, if Q is lower than Qt, then q
• must be higher than one.

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• There are two reasons could cause this
• problem. One is that the model is too small,
• and we cannot get the answer to this problem.
• Alternatively is because q could be any non-
• negative number in our model. If we want to
• test which reason is right, we need a larger
• sample.

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• (c) From your knowledge of the theory of omitted
  variables, what does this omission do to your
  estimate of the models parameters?
• We can get the answer from Part(b) obviously.

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Problem 7

• Using the data provided, estimate equation (8)
  using ordinary least squares.
• log(St) ab1log(Pt-1)b2log(PC,t-1)
• b3log(PB,t-1)b4log(Wt-1)et
• 
  (8)

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SHAZAM Command

• SAMPLE 2 16
• READ(D/123.TXT) S PLAG PCLAG PBLAG WLAG
• GENR lnSLOG(S)
• GENR lnPLOG(PLAG)
• GENR lnPCLOG(PCLAG)
• GENR lnPBLOG(PBLAG)
• GENR lnWLOG(WLAG)
• GENR lnQLAG(lnS)
• OLS lnS lnP lnPC lnPB lnW

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SHAZAM OUTPUT
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• (b) Test your model for the existence of a
  regression at the 5 level of significance. Does
  it appear that you have a regression?
• We can use anf command to compute the
• F-value to see the significance.

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• The 10 level of significance for this
• condition is F3.48gt0.9154272. So we reject
• Ho, and our model is insignificance under a
• 10 level. So it appears that we should not
• reject our hypothesis, and we have got our
• regression.

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Problem 8

• (a) Test the model estimated in Problem (7) for
  autocorrelation at the 5 level of significance
  using the Durbin-Watson test.

42
SHAZAM Command

• SAMPLE 2 16
• READ(D/123.TXT) S PLAG PCLAG PBLAG WLAG
• GENR lnSLOG(S)
• GENR lnPLOG(PLAG)
• GENR lnPCLOG(PCLAG)
• GENR lnPBLOG(PBLAG)
• GENR lnWLOG(WLAG)
• GENR lnQLAG(lnS)
• OLS lnS lnP lnPC lnPB lnW/DWPVALUE

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SHAZAM OUTPUT
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• From the output, we can see that the P-value
• are both higher than 5, so there is no
• autocorrelation.

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• (b) Are we able to reach a decision about the
  presence of autocorrelation given the results of
  your Durbin-Watson test?
• Yes, since there is no more lagged
• dependent variables as one of the
• independent variables. So we can use the
• classical Durbin-Watson test.

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• (c) In Problem (2) we were unable to utilize the
  Durbin-Watson test. Why can we now use this test?
  
• It is because that in Problem 2, there is a
• lagged variable in the model, which is
• lnQ(lnS(1.1)), while there is no more lagged
• variable in equation (8) in Problem 7.
• Therefore we can use the classical Durbin-
• Watson test now.

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Problem 9

• (a) If you were unable to reach a decision about
  autocorrelation in Problem (8) while using the
  Durbin-Watson test, test the model estimated in
  Problem (7) for first order autocorrelation at
  the 5 level of significance using the Lagrange
  Multiplier test.

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SHAZAM Command

• SAMPLE 2 16
• READ(D/123.TXT) S PLAG PCLAG PBLAG WLAG
• GENR lnSLOG(S)
• GENR lnPLOG(PLAG)
• GENR lnPCLOG(PCLAG)
• GENR lnPBLOG(PBLAG)
• GENR lnWLOG(WLAG)
• GENR lnQLAG(lnS)
• OLS lnS lnP lnPC lnPB lnW

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SHAZAM OUTPUT
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• First we should calculate the Fitted value
  fvln(s)
• Then we regression the function
• etyb4et-1u where
  yfvln(s)

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• The SHAZAM output

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• LM Statistics(LM(c2)
• (n-p)R2(15-1)0.11321.5848
• The Chi-squared is 3.84146 at the 5 level,
• while our chi-squared value is 1.5848, which is
• lower than 3.84146.
• Therefore, there is no evidence of
• autocorrelation.

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• Since we get the answer of that there is no
• evidence of autocorrelation. So we do not
• need to consider Problem (9)-(c), (d) and
• Problem (10).

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Congratulations

Weve done!