heteroscedasticity 1 this sequence relates to assumption a.4 of the regression model assumptions and...
DESCRIPTION
3 If there were no disturbance term in the model, the observations would lie on the line as shown. HETEROSCEDASTICITY 11 X Y = 1 + 2 X Y X3X3 X5X5 X4X4 X1X1 X2X2TRANSCRIPT
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This sequence relates to Assumption A.4 of the regression model assumptions and introduces the topic of heteroscedasticity. This relates to the distribution of the disturbance term in a regression model.
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Y = 1 +2X
Y
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Y = 1 +2X
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We will discuss it in the context of the regression model Y = 1 + 2X + u. To keep the diagram uncluttered, we will suppose that we have a sample of only five observations, the X values of which are as shown.
X3 X5X4X1 X2
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If there were no disturbance term in the model, the observations would lie on the line as shown.
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X
Y = 1 +2X
Y
X3 X5X4X1 X2
1
X
Y = 1 +2X
Y
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Now we take account of the effect of the disturbance term. It will displace each observation in the vertical dimension, since it modifies the value of Y without affecting X.
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The disturbance term in each observation is hypothesized to be drawn randomly from a given distribution. In the diagram, three assumptions are being made.
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Y = 1 +2X
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One is that the expected value of u in each observation is 0 (Assumption A.3). The second is that the distribution in each observation is normal (Assumption A.6). We are not concerned with either of these and we will assume them to be true.
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Y = 1 +2X
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The third, Assumption A.4, is that the variance of the distribution of the disturbance term is the same for each observation. In the present case, that means that the normal distributions shown all have the same variance.
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Y = 1 +2X
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If Assumption A.4 is satisfied, the disturbance term is said to be homoscedastic (Greek for same scattering).
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Y = 1 +2X
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Each observation is then potentially (before the sample is drawn) an equally reliable guide to the location of the line Y = 1 + 2X.
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Y = 1 +2X
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Once the sample has been drawn, some observations will lie closer to the line than others, but we have no way of anticipating in advance which ones these will be.
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Y = 1 +2X
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Now consider the situation illustrated by the diagram above. The distribution of u associated with each observation still has expected value 0 and is normal. However Assumption A.4 is violated and the variance is no longer constant.
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Y = 1 +2X
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Obviously, observations where u has low variance, like that for X1, will tend to be better guides to the underlying relationship than those like that for X5, where it has a relatively high variance.
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When the distribution is not the same for each observation, the disturbance term is said to be subject to heteroscedasticity.
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There are two major consequences of heteroscedasticity. One is that the standard errors of the regression coefficients are estimated wrongly and the t tests (and F test) are invalid.
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The other is that OLS is an inefficient estimation technique. An alternative technique which gives relatively high weight to the relatively low-variance observations should tend to yield more accurate estimates.
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Y = 1 +2X
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In the scatter diagram manufacturing output is plotted against GDP, both measured in US$ million, for 30 countries for 1997. The data are from the UNIDO Yearbook. The sample is restricted to countries with GDP at least $10 billion and GDP per capita at least $2000.
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The scatter diagram is dominated by the observations for Japan and the USA and it is difficult to detect any kind of pattern.
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However it those two countries are dropped and the scatter diagram rescaled, a clear picture of heteroscedasticity emerges.
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The reason for the heteroscedasticity is that variations in the size of the manufacturing sector around the trend relationship increase with the size of GDP.
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South Korea and Mexico are both countries with relatively large GDP. The manufacturing sector is relatively important in South Korea, so its observation is far above the trend line. The opposite was the case for Mexico, at least in 1997.
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Singapore and Greece are another pair of countries with relatively large and small manufacturing sectors. However, because the GDP of both countries is small, their variations from the trend relationship are also small.
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Copyright Christopher Dougherty 2012.
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The content of this slideshow comes from Section 7.1 of C. Dougherty, Introduction to Econometrics, fourth edition 2011, Oxford University Press.Additional (free) resources for both students and instructors may be downloaded from the OUP Online Resource Centrehttp://www.oup.com/uk/orc/bin/9780199567089/.
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2012.11.10