Solutions to Problem Set 2
Inflation and Central Bank Independence
We start by reading the data as instructed
. use http://data.princeton.edu/wws509/datasets/inflation (Inflation and central bank independence)
[1] The Inflation Rate
(a) Plot the inflation rate versus the questionnaire-based measure of central bank independence, using different symbols for developed and developing countries. Make sure you identify the point on the lower left as Ethiopia. (In part d you will add to this plot; turn in only the final result.)
Here's the final plot:
. rename ques cbi // I prefer the name cbi to ques
. gen ET = cbi < 1.5 // Ethiopia has the lowest cbi!
. twoway (scatter inf cbi if dev, msymbol(square)) ///
> (scatter inf cbi if !dev, msymbol(triangle)) ///
> (scatter inf cbi if ET, msymbol(none) mlabel(country)) ///
> (lfit inf cbi if dev) (lfit inf cbi if !dev & !ET) ///
> , legend(ring(0) pos(1) order(1 "Developed" 2 "Not Developed")) ///
> title("Inflation and Central Bank Independence in 22 Countries")
. graph export ps2fig1.png, width(400) replace
(file ps2fig1.png written in PNG format)
We see that inflation increases with lack of central bank independence in developing countries and is comparatively flat in develop countries. Ethiopia is unsual in having the lowest measure of central bank independence and practically no inflation.
For simplicity, exclude Ethiopia from the analyses in parts 1.b, 1.c and 1.d
(b) Regress the inflation rate on the indicator variable for developed countries and interpret the coefficients. Test the significance of the coefficient of developed countries.
. reg inf dev if !ET
Source | SS df MS Number of obs = 21
-------------+------------------------------ F( 1, 19) = 14.68
Model | 7230.98225 1 7230.98225 Prob > F = 0.0011
Residual | 9360.82727 19 492.67512 R-squared = 0.4358
-------------+------------------------------ Adj R-squared = 0.4061
Total | 16591.8095 20 829.590476 Root MSE = 22.196
------------------------------------------------------------------------------
inf | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
dev | -37.15455 9.698255 -3.83 0.001 -57.45323 -16.85586
_cons | 44.7 7.019082 6.37 0.000 30.00889 59.39111
------------------------------------------------------------------------------
We see that the inflation rate in developed countries is 37.2 percentage points lower than in developing countries. The difference is clearly significant with a t-test of -3.83 on 19 d.f.(equivalent to an F-test of 14.68 on 1 and 19 d.f.). [Obviously including Ethiopia would reduce this difference.]
(c) Add the measure of central bank independence to the model and comment on the results. What happens to the difference between developed and developing countries once you adjust for central bank independence?
. reg inf dev cbi if !ET
Source | SS df MS Number of obs = 21
-------------+------------------------------ F( 2, 18) = 15.07
Model | 10387.9256 2 5193.96282 Prob > F = 0.0001
Residual | 6203.88388 18 344.660215 R-squared = 0.6261
-------------+------------------------------ Adj R-squared = 0.5845
Total | 16591.8095 20 829.590476 Root MSE = 18.565
------------------------------------------------------------------------------
inf | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
dev | -21.60823 9.601308 -2.25 0.037 -41.77983 -1.436628
cbi | -7.634352 2.522519 -3.03 0.007 -12.93397 -2.334737
_cons | 82.10833 13.68371 6.00 0.000 53.35991 110.8567
------------------------------------------------------------------------------
The results indicate that the inflation rate in developed countries is 21.6 percentage points lower than in developing countries with the same level of central bank independence. This difference is still significant after adjusting, but it has been reduced in magnitude indicating that part of the differences between developed and developing countries can be attributed to differences in central bank independence. [Including Ethiopia would reduce the adjusted difference and in fact render it non-significant.]
(d) Test whether the slope of inflation by central bank independence is the same in developed and developing countries by adding an interaction effect. Superimpose the fitted lines from this model on the plot of part a.
. gen devXcbi = dev*(cbi-6)
. reg inf dev cbi devXcbi if !ET
Source | SS df MS Number of obs = 21
-------------+------------------------------ F( 3, 17) = 13.73
Model | 11743.5947 3 3914.53157 Prob > F = 0.0001
Residual | 4848.21483 17 285.189107 R-squared = 0.7078
-------------+------------------------------ Adj R-squared = 0.6562
Total | 16591.8095 20 829.590476 Root MSE = 16.888
------------------------------------------------------------------------------
inf | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
dev | -24.25895 8.817976 -2.75 0.014 -42.86325 -5.654646
cbi | -11.19989 2.817723 -3.97 0.001 -17.14476 -5.255012
devXcbi | 10.58504 4.854921 2.18 0.044 .3420529 20.82803
_cons | 99.57945 14.80364 6.73 0.000 68.3465 130.8124
------------------------------------------------------------------------------
The first thing to note is that the interaction term is significant at the conventional
5% level, so we reject the hypothesis that the slopes are the same. In developing
countries the slope is -11.2, indicating that inflation is lower in countries where
the central bank has more independence, at the rate of 11.2 percentage points less
inflation per point in the central bank independence measure. In developed countries
the slope is -0.6 (calculated as -11.2 + 10.6), suggesting that differences by central
bank independence are much smaller than in developing countries. In fact, we have no
evidence that the line for developed countries is not flat (test not shown). Because I
centered cbi on 6, a value very close to the median, when computing the
interaction, the coefficient of dev tells us that the inflation rate is
24.2 percentage points lower in developed than developing countries at the median level
of central bank independence. The lines requested were included in the graph shown above.
(e) Comment briefly on how the conclusions of the analysis in part 2.d would be altered if we included Ethiopia.
Obviously including Ethiopia would pull the line would developing countries down at the the low independence end, making it less step. This would reduce the estimated difference in slopes, in fact making it non significant:
. reg inf dev cbi devXcbi
Source | SS df MS Number of obs = 22
-------------+------------------------------ F( 3, 18) = 4.93
Model | 7680.60602 3 2560.20201 Prob > F = 0.0113
Residual | 9341.75762 18 518.986534 R-squared = 0.4512
-------------+------------------------------ Adj R-squared = 0.3597
Total | 17022.3636 21 810.588745 Root MSE = 22.781
------------------------------------------------------------------------------
inf | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
dev | -24.82716 11.89387 -2.09 0.051 -49.81526 .1609369
cbi | -5.641294 3.298455 -1.71 0.104 -12.57109 1.288503
devXcbi | 5.026446 6.270925 0.80 0.433 -8.148279 18.20117
_cons | 66.7961 16.57334 4.03 0.001 31.97681 101.6154
------------------------------------------------------------------------------
Consider this a cautionary tale of how much the results can change if we include an observation that obviously does not follow the same model as the others.
[2] Working with Log(Inflation)
(a) Plot the log of the inflation rate versus the measure of central bank independence. (In part d you will add to this plot. Turn in only the final result.)
Here's my plot, with a addition of two other countries that proved useful in part 3.
. gen CR = country == "Costa Rica"
. gen DE = country == "Germany"
. gen pos = 3
. replace pos = 9 if DE
(1 real change made)
. gen linf = log(inf)
(1 missing value generated)
. twoway (scatter linf cbi if dev, msymbol(circle)) ///
> (scatter linf cbi if !dev, msymbol(triangle)) ///
> (scatter linf cbi if ET | CR | DE, msymbol(none) mlabel(country) mlabv(pos)) ///
> (lfit linf cbi if dev) (lfit linf cbi if !dev & !ET) ///
> , legend(ring(0) pos(1) order(1 "Developed" 2 "Not Developed")) ///
> title("Log-Inflation and Central Bank Independence in 22 Countries")
. graph export ps2fig2.png, width(400) replace
(file ps2fig2.png written in PNG format)
Working with the log of the inflation rate confirms the impressions we had earlier but shows more clearly what's going on at the low inflation end, where differences by central bank independence become clearer.
For simplicity, exclude Ethiopia from the analyses in parts 2.b, 2.c and 2.d
(b) Regress the log of the inflation rate on the indicator variable for developed countries and interpret the coefficient of developed countries. (Be careful if your interpretation relies on a common approximation; you should make sure it is reasonable in this case.)
. reg linf dev if !ET
Source | SS df MS Number of obs = 21
-------------+------------------------------ F( 1, 19) = 23.07
Model | 12.2253436 1 12.2253436 Prob > F = 0.0001
Residual | 10.0705669 19 .530029838 R-squared = 0.5483
-------------+------------------------------ Adj R-squared = 0.5245
Total | 22.2959105 20 1.11479553 Root MSE = .72803
------------------------------------------------------------------------------
linf | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
dev | -1.52772 .3180998 -4.80 0.000 -2.193511 -.8619296
_cons | 3.474515 .2302238 15.09 0.000 2.992652 3.956379
------------------------------------------------------------------------------
. di exp(_b[dev])-1
-.78297013
The coefficient of -1.53 tells us that the inflation rate is 78.3% lower in developed than in developing countries. (This is not a small coefficient, so e^b-1 is not ~ b, a conclusion that the inflation rate is 153% lower would be absurd.)
(c) Add the measure of central bank independence to the model and interpret the coefficients.
. reg linf dev cbi if !ET
Source | SS df MS Number of obs = 21
-------------+------------------------------ F( 2, 18) = 19.45
Model | 15.2435574 2 7.62177868 Prob > F = 0.0000
Residual | 7.05235315 18 .391797397 R-squared = 0.6837
-------------+------------------------------ Adj R-squared = 0.6485
Total | 22.2959105 20 1.11479553 Root MSE = .62594
------------------------------------------------------------------------------
linf | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
dev | -1.047026 .3237171 -3.23 0.005 -1.72713 -.3669215
cbi | -.2360553 .0850491 -2.78 0.012 -.4147368 -.0573739
_cons | 4.631187 .4613592 10.04 0.000 3.661907 5.600466
------------------------------------------------------------------------------
. di exp(_b[dev])-1
-.64901992
. di exp(_b[cbi])-1
-.21026301
The inflation rate in developed countries is 64.9% lower than in developing countries with the same level of central bank independence. Each additional point in the central bank independence scale is associated with a 21% reduction in the inflation rate. (The coefficient of -0.236 is almost small enough in magnitude to be interpreted directly as a percent reduction.)
(d) Test for an interaction between central bank independence and the indicator of development. Superimpose the fitted lines from this model on the plot of part 2.a. Do we have evidence that these lines are not parallel?
. reg linf dev cbi devXcbi if !ET
Source | SS df MS Number of obs = 21
-------------+------------------------------ F( 3, 17) = 13.27
Model | 15.6246221 3 5.20820736 Prob > F = 0.0001
Residual | 6.67128842 17 .392428731 R-squared = 0.7008
-------------+------------------------------ Adj R-squared = 0.6480
Total | 22.2959105 20 1.11479553 Root MSE = .62644
------------------------------------------------------------------------------
linf | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
dev | -1.091467 .3271017 -3.34 0.004 -1.781591 -.4013428
cbi | -.2958341 .1045231 -2.83 0.012 -.5163586 -.0753097
devXcbi | .177466 .1800927 0.99 0.338 -.2024964 .5574283
_cons | 4.924103 .5491393 8.97 0.000 3.76552 6.082685
------------------------------------------------------------------------------
. di exp(_b[dev])-1
-.6642764
. di exp(_b[cbi])-1
-.2560892
. di exp(_b[devXcbi])-1
.1941874
Interestingly the interaction in this model is not significant; in other words we have no evidence that differences in the inflation rate between developed and developing countries are relatively larger at lower levels of central bank independence. By the same token the relative effect of central bank independence appears to be the same in developed and developed countries.
This example illustrate sthe fact that interactions are always specific to a given scale. Two variables that interact when we look at absolute differences in inflation no longer interaction when we look at relative differences, resulting in a simpler model.
(e) Comment briefly on how including Ethiopia would alter your conclusions in part 2.d.
It looks form the plot that including or excluding Ethiopia would still make a difference, but perhaps a smaller one than when we work in the log scale
. reg linf dev cbi devXcbi
Source | SS df MS Number of obs = 22
-------------+------------------------------ F( 3, 18) = 4.63
Model | 10.4017126 3 3.46723755 Prob > F = 0.0143
Residual | 13.4777026 18 .748761257 R-squared = 0.4356
-------------+------------------------------ Adj R-squared = 0.3415
Total | 23.8794153 21 1.13711501 Root MSE = .86531
------------------------------------------------------------------------------
linf | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
dev | -1.113581 .4517695 -2.46 0.024 -2.062714 -.164449
cbi | -.0794977 .1252865 -0.63 0.534 -.3427148 .1837194
devXcbi | -.0388705 .238191 -0.16 0.872 -.5392911 .4615501
_cons | 3.648198 .6295114 5.80 0.000 2.325644 4.970753
------------------------------------------------------------------------------
. di exp(_b[dev])-1
-.67161923
. di exp(_b[cbi])-1
-.07641984
. di exp(_b[devXcbi])-1
-.03812474
The results show that this is indeed the case, the line for developing countries is less step than before, but the difference between develop and developing countries at the median level of central bacnk independence is still significant, whereas it just failed the 5% level when we worked with the inflation rate.
[3] Regression Diagnostics
Calculate the following diagnostics for the additive model of part 2.c including Ethiopia, so we can see what the different measures would say about this country.
(a) Compute leverages and comment briefly on the four countries with the most leverage. Why do you think Costa Rica comes at the top of the list?
. reg linf dev cbi
Source | SS df MS Number of obs = 22
-------------+------------------------------ F( 2, 19) = 7.31
Model | 10.3817723 2 5.19088614 Prob > F = 0.0044
Residual | 13.497643 19 .710402263 R-squared = 0.4348
-------------+------------------------------ Adj R-squared = 0.3753
Total | 23.8794153 21 1.13711501 Root MSE = .84285
------------------------------------------------------------------------------
linf | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
dev | -1.124559 .4351395 -2.58 0.018 -2.035317 -.213802
cbi | -.0902519 .1037896 -0.87 0.395 -.307486 .1269822
_cons | 3.697374 .5383573 6.87 0.000 2.57058 4.824169
------------------------------------------------------------------------------
. predict plinf
(option xb assumed; fitted values)
. gen pinf = exp(plinf)
. predict lev, lev
. gsort -lev // sort in descending order
. list country dev cbi inf pinf lev in 1/5
+----------------------------------------------------+
| country dev cbi inf pinf lev |
|----------------------------------------------------|
1. | Costa Rica 0 8.1 23 19.4206 .2795697 |
2. | Ethiopia 0 1.3 4 35.87523 .2533228 |
3. | Germany 1 10 3 5.313742 .2332329 |
4. | Yugoslavia 0 1.7 73 34.6032 .2160479 |
5. | Bahamas 0 7.1 6 21.25487 .1877608 |
+----------------------------------------------------+
Interestingly enough Costa Rica turned out to have more leverage than Ethiopia.
Recall that leverage is a function of just the predictors. Ethiopia has some leverage
because it has a very low cbi score, but so do two other developing countries.
Costa Rica is unusual in having a very high cbi score among developing
countries, with no other developing country coming close.
(b) Calculate the jack-knifed residuals and list any countries with values in excess of 2. Do we have evidence of any outliers, after making allowance for the fact that we are making multiple tests?
. predict rstu, rstu
(1 missing value generated)
. gen absrstu = abs(rstu)
(1 missing value generated)
. gsort -absrstu
. list country dev cbi inf pinf rstu in 1/5
+---------------------------------------------------+
| country dev cbi inf pinf rstu |
|---------------------------------------------------|
1. | Ethiopia 0 1.3 4 35.87523 -4.05593 |
2. | Bahamas 0 7.1 6 21.25487 -1.753668 |
3. | Barbados 0 5.4 7 24.77944 -1.652541 |
4. | Peru 0 2.2 108 33.07641 1.610441 |
5. | Uganda 0 5.3 72 25.00409 1.350238 |
+---------------------------------------------------+
The only residual exceeding 2 in absolute value is Ethiopia. With 22 countries the Bonferroni critical value is
. di invttail(18, 0.025/22) 3.5530015
because the residual for Ethiopia exceeds 3.55 in absolute value we can conclude with 95% confidence that this country is an outlier; in other words, it does not follow the same model as the rest.
(c) Compute the Cook distances. Any indication of an influential observation? Why is Ethiopia at the low end of the scale of central bank independence a lot more influential than Germany at the high end?
. predict cook, cook
(1 missing value generated)
. gsort -cook
. list country dev cbi inf pinf lev rstu cook if _n < 5 | DE
+-------------------------------------------------------------------------+
| country dev cbi inf pinf lev rstu cook |
|-------------------------------------------------------------------------|
1. | Ethiopia 0 1.3 4 35.87523 .2533228 -4.05593 1.026026 |
2. | Bahamas 0 7.1 6 21.25487 .1877608 -1.753668 .2136355 |
3. | Peru 0 2.2 108 33.07641 .1762778 1.610441 .1706901 |
4. | Barbados 0 5.4 7 24.77944 .1012868 -1.652541 .0940266 |
7. | Germany 1 10 3 5.313742 .2332329 -.7661245 .0608344 |
+-------------------------------------------------------------------------+
The only country to exceed the cutoff of one, indicating that it had substantial incluence in the results, is Ethiopia, so we have done well to exclude it.
The reason Ethiopia has such large influence is that it combines high leverage with a large residual; in other words it had potential influence and had an unexpected outcome. In contrast, Germany had almost equal leverage but the residual was modest, so its inflation rate behaved like that of other countries.
[4] Box-Cox Transformations
(a) Did we need to transform the data? Was the natural logarithm a good transformation? Explore these two questions in a Box-Cox framework by formally testing for the log and identity transformations using an additive model excluding Ethiopia.
This is easy to do with the box-cox commnand. Note that we specify the variable in the original scale
. boxcox inf dev cbi if !ET, model(lhs)
Fitting comparison model
Iteration 0: log likelihood = -99.8552
Iteration 1: log likelihood = -86.423172
Iteration 2: log likelihood = -85.698169
Iteration 3: log likelihood = -85.697791
Iteration 4: log likelihood = -85.697791
Fitting full model
Iteration 0: log likelihood = -89.525997
Iteration 1: log likelihood = -78.135069
Iteration 2: log likelihood = -74.404917
Iteration 3: log likelihood = -74.139808
Iteration 4: log likelihood = -74.139759
Iteration 5: log likelihood = -74.139759
Number of obs = 21
LR chi2(2) = 23.12
Log likelihood = -74.139759 Prob > chi2 = 0.000
------------------------------------------------------------------------------
inf | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
/theta | -.1758896 .2086666 -0.84 0.399 -.5848686 .2330895
------------------------------------------------------------------------------
Estimates of scale-variant parameters
----------------------------
| Coef.
-------------+--------------
Notrans |
dev | -.6350744
cbi | -.1396188
_cons | 3.242201
-------------+--------------
/sigma | .3558902
----------------------------
---------------------------------------------------------
Test Restricted LR statistic P-value
H0: log likelihood chi2 Prob > chi2
---------------------------------------------------------
theta = -1 -80.932391 13.59 0.000
theta = 0 -74.500419 0.72 0.396
theta = 1 -89.525997 30.77 0.000
---------------------------------------------------------
The table at the bottom of the output tells us that we reject the hypothesis that the transformation parameter is 1 (indicating no need to transform), with a chi-squared of 31 on 1 df. The log transformation, corresponding to a transformation parameter of 0, cannot be rejected, with a chi-squared of 0.72 on one df, and in fact is very close to the optimum transformation of -0.18 (which would be a pain to interpret).
(b) Plot the Box-Cox profile log-likelihood as a function of the transformation parameter. The Stata logs section of the website shows how to do this in a simple loop. log close
. scalar maxlogL = e(ll) // save max of logL
. gen logL = .
(23 missing values generated)
. gen theta = .
(23 missing values generated)
. local i = 1
. forvalues theta = -1(.1)1 {
2. quietly {
3. boxcox inf dev cbi if !ET, model(lhs) from(`theta',copy) iterate(0)
4. replace theta = `theta' in `i'
5. replace logL = e(ll) in `i'
6. local i = `i' + 1
7. }
8. }
. gen cb = maxlogL - 3.84/2 if abs(theta) < .75
(8 missing values generated)
. graph twoway (line logL theta) (line cb theta), ///
> title("Box-Cox Profile Log-Likelihood") ///
> xtitle(theta) ytitle(log-likelihood) legend(off)
. graph export ps2fig3.png, width(400) replace
(file ps2fig3.png written in PNG format)
We see that any transformation between -.5 and .2 would be acceptable, that the log transformation is close to the optimum, and that leaving the data untransformed is soundly rejected.
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