ࡱ> 02/w ?!bjbj 3LjjSl$$$8\,tr"!!!!!!!$\# |%!-!,"!! ! 0Z$4B !B"0r" ,F&vF&!Testing for Assumptions (listed in 6.2) of the Disturbance of the Population Regression Assumption  Plot (look for key patterns) Statistic (summarize key patterns) Linearity E( e | Xs) = 0  Residual vs. Predicted  NA Constant Variance Var(e | Xs) = s2  Residual vs. Predicted Breusch - Pagan Normality  Histogram Normal plot (Q-Q or P-P) (See Appendix I)  Skewness Kurtosis Jarque  Bera (See Appendix II)  Independence  Line plot of the residual  Durbin - Watson Appendix I: Testing for Normality By Using a Q-Q Plot A natural question in applying a normal distribution is: how can we test whether the the data actually come from a normal distribution? A simple method is to construct a histogram, and compare the shape with the normal distribution that has the same mean and the standard deviation as the sample mean and the sample standard deviation of the data, respectively. Fortunately, several convenient statistical packages are available for drawing both the histogram with the normal curve superimposed. As an example, the figure below shows a histogram with a normal curve for recent 61 observations of the monthly stock rate of return of Exxon.  The histogram uses only 61 observations, whereas the normal curve superimposed depicts the histogram using infinitely many observations. Therefore, sampling errors should show up as gaps between the two curves. The graph shows that the distribution of the rate of return of Exxon appears not markedly different from a normal distribution at least in the middle part. The procedure described above is easy to understand, but is not effective for revealing a subtle but systematic departure of the histogram from normality. A better graphical check of normality is a normal probability plot. The plot can be easily developed using Excel and we describe the process in below. The first step is to sort the data from the lowest to the highest. Let n be the number of observations. Then, the lowest observation, denoted as x(1) is the (1/n) th quantile of the data. A quantile times 100 is the percentile, so x(1) is also the (1/n) x 100 th percentile of the data. With this convention, however, the largest observation becomes the 100 percentile of the data, which presents a problem as the 100 percentile of a normal distribution is infinity, the value that can never be assumed in observation. A suggested choice is to define the i-th largest observation, x(i) as the (i/(n+1))th quantile, or the (i/(n+1)) x 100 percentile of the data. In the Excel worksheet on the next page both choices are computed for comparison. The next step is to determine for each observation the corresponding quantile of the normal distribution that has the same mean and the standard deviation as the data. The following Excel function is a convenient way to determine the normal (i/(n+1)) th quantile, denoted as x(i). x(i) = NORMINV(i/(n+1), sample mean, sample standard deviation).  EMBED Word.Picture.6  i/(n+1) x(i) This value is the expected quantile if the data come from a normal distribution. x(i) should be close to x(i) if the normality of the distribution is true. The quantile of the normal distribution -- with the mean and the standard deviation equaling the sample mean and the sample standard deviation, respectively -- are computed in the column with the heading Expected. A normal probability plot is a scatterplot of the data vs. the expected quantiles. the plot is shown below. If the data indeed come from a normal distribution, then the scatterplot should deviate in a random fashion from the reference line. Note that the 45 degree line serves as a convenient reference line for detecting a systematic departure from normality.  EMBED Excel.Chart.5 \s  We show below two normal probability plots, one for a sample of 60 independent observations from a uniform distribution over the interval (0, 1) and the other for a sample of 60 independent observations from an exponential distribution with mean 0.5. Probabilities at both tails of a uniform distribution is significantly larger than a normal distribution. On the other hand, an exponential distribution is skewed. These departures from the normality are quite common and the two plots show how each departure can be detected by using the normal probability plots.  Normal Probability Plot of Data From a Uniform Distribution The plot on the right is a normal probability plot of observations from a uniform distribution. The plot has an elongated S shape. Normal Probability Plot of Data From an Exponential Distribution  The plot on the right is a normal probability plot of observations from an exponential distribution. The plot is convex. Appendix II: Testing for Normality By Using a Jarque-Bera Statistic A normal probability plot test can be inconclusive when the plot pattern is not clear. In such case it is useful to compute a few numbers that measure non-normality. The asymmetry of the distribution is measured by the skewness which is the third central moment of the distribution:  EMBED Equation.2  The sample skewness is evaluated as follows:  EMBED Equation.2  where:  EMBED Equation.2  The skewness  EMBED Equation.2 is 0 for a symmetric population, as can been seen from the formula. Therefore, the sample skewness  EMBED Equation.2  is significantly different from 0, then one can infer that the population distribution is unlikely to be symmetric and hence not normal. Another number that can be used to check the normality of the distribution is the the fourth central moment of the distribution, called the kurtosis  EMBED Equation.2 :  EMBED Equation.2  The sample kurtosis is computed as follows:  EMBED Equation.2  The kurtosis measures the amount of the tail probabilities of the distribution and equals 3 for a normal population distribution. Therefore, if the sample kurtosis  EMBED Equation.2  is significantly different from 3, then one can infer that the population distribution is unlikelty to be not normal. The Jarque-Bera Statistic combines the two measures  EMBED Equation.2  and EMBED Equation.2  as follows:  EMBED Equation.2  EMBED Equation.2  For a large number of observations, JB higher than 6 suggests that the population distribution is unlikely to be normal. The worksheet below illustrates the computation of these statistics for the rate of return of the Weyerhaeuser stock for recent 60 months. Computing the Skewness, the Kurtosis and the Jarque-Bera Statistc for Recent 60 Monthly Return of the Weyerhaeuser Stock  <oZ\^`+ . 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PSymbol |w|wgw -!a Times U|w|wgw U -!3 & ' )dxpr  " currentpoint ",Times .+  , Symbol( a  +3/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 448 div 512 3 -1 roll exch div scale currentpoint translate 64 48 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def /mt_vec StandardEncoding 256 array copy def /Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute /agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave /ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde/oacute /ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex/udieresis mt_vec 128 32 getinterval astore pop mt_vec dup 176 /brokenbar put dup 180 /twosuperior put dup 181 /threesuperior put dup 188 /onequarter put dup 190 /threequarters put dup 192 /Agrave put dup 201 /onehalf put dup 204 /Igrave put pop /Egrave/Ograve/Oacute/Ocircumflex/Otilde/.notdef/Ydieresis/ydieresis /Ugrave/Uacute/Ucircumflex/.notdef/Yacute/thorn mt_vec 209 14 getinterval astore pop mt_vec dup 228 /Atilde put dup 229 /Acircumflex put dup 230 /Ecircumflex put dup 231 /Aacute put dup 236 /Icircumflex put dup 237 /Iacute put dup 238 /Edieresis put dup 239 /Idieresis put dup 253 /yacute put dup 254 /Thorn put pop /re_dict 4 dict def /ref { re_dict begin /newfontname exch def /basefontname exch def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict { exch dup dup /FID ne exch /Encoding ne and { exch newfont 3 1 roll put } { pop pop } ifelse } forall newfont /FontName newfontname put newfont /Encoding mt_vec put newfontname newfont definefont pop end } def /Times-Roman /MT_Times-Roman ref 384 /MT_Times-Roman f1 (\303) 73 258 sh /f2 {ff matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (a) -30 304 sh 224 /MT_Times-Roman f1 (3) 237 400 sh end MTsave restore d'MATH "a  3 dFMicrosoft Equation Editor 2.0DNQE Equation.2ObjInfo;=gEquation Native h7_944647620.N@Fy0y0Ole i "a  3drrrr    .  & PSymbol ||w|wgw | -!a Times /|w|wgw / PIC ?BjdMETA l@PICT ADqCompObjY-!4 & 'dxpr  " currentpoint ", Symbol .+ a,Times +4/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 448 div 448 3 -1 roll exch div scale currentpoint translate 64 59 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def /f2 {ff matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (a) -30 261 sh 224 /Times-Roman f1 (4) 244 357 sh end MTsave restore d$MATH a 4FMicrosoft Equation Editor 2.0DNQE Equation.2 a 4dzOhzO    R .ObjInfoCEEquation Native 4_944647669HF00Ole PIC GJdMETA PICT ILCompObjY  & PSymbol |w|wgw  -!aTimes8 |w|wgw8 -!4 PSymbol |w|wgw  -!=Times8 |w|wgw8 -!E!x ,PSymbol |w|wgw -!- 4PSymbol8 |w|wgw8 -!m >!s4-+EPSymbol !|w|wgw ! -! %! %! %! F! F! FTimes8 |w|wgw8 -!4M & ' Rdxpr  R" R currentpoint ", Symbol .+a,Times +4 (=) E( ,x)-) m(4s"+( % * (% ( F * (F  (M4/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2624 div 1024 3 -1 roll exch div scale currentpoint translate 64 41 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 1324 500 moveto 866 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def /f2 {ff matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (a) -30 599 sh (m) 1933 350 sh (s) 1609 900 sh 224 /Times-Roman f1 (4) 244 695 sh (4) 2402 152 sh 384 /Symbol f1 (=) 569 599 sh (-) 1631 350 sh (\346) 1127 412 sh (\350) 1127 831 sh (\366) 2200 412 sh (\370) 2200 831 sh 384 /Times-Italic f1 (E) 845 599 sh (x) 1367 350 sh end MTsave restore dbMATHV  a 4 =Ex-ms() 4FMicrosoft Equation Editor 2.0DNQE Equation.2ObjInfoKMEquation Native r_944647891F^PF00Ole V a 4 =Ex-ms() 4d l    "_ .  & PIC ORdMETA HPICT QT CompObjYTimesO |w|wgwO -! PSymbol ||w|wgw | -!a TimesO |w|wgwO -!4 PSymbol }|w|wgw } -!=TimesO |w|wgwO -!1 Times ~|w|wgw ~ -!n-"!x 9TimesO |w|wgwO -!i?PSymbol |w|wgw  -!- DMQTimesO |w|wgwO -!x LTimes |w|wgw -! DPSymbolO |w|wgwO -!s A8RPSymbol |w|wgw -! 2! 2! 2! S! S! STimesO |w|wgwO -!i %PSymbol |w|wgw -!= (TimesO |w|wgwO -!1 ,Times |w|wgw -!n)PSymbolO |w|wgwO -!$Times |w|wgw -!4Z & ' "_dxpr  "_""_ currentpoint ",Times .+ , Symbol(a  +4 (=( 1*n"( 9x +i ( D-"M)x (D   !"#$%&'()*+,-./0123456789:;=BDEFGHIKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnoqvxyz{|} (As "8(2 * (2 (S * (S  ( %i)=)1()n ($ (Z4G/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 3040 div 1088 3 -1 roll exch div scale currentpoint translate 64 61 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 839 512 moveto 236 0 rlineto stroke 2408 119 moveto 170 0 rlineto stroke 1745 512 moveto 865 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def /mt_vec StandardEncoding 256 array copy def /Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute /agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave /ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde/oacute /ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex/udieresis mt_vec 128 32 getinterval astore pop mt_vec dup 176 /brokenbar put dup 180 /twosuperior put dup 181 /threesuperior put dup 188 /onequarter put dup 190 /threequarters put dup 192 /Agrave put dup 201 /onehalf put dup 204 /Igrave put pop /Egrave/Ograve/Oacute/Ocircumflex/Otilde/.notdef/Ydieresis/ydieresis /Ugrave/Uacute/Ucircumflex/.notdef/Yacute/thorn mt_vec 209 14 getinterval astore pop mt_vec dup 228 /Atilde put dup 229 /Acircumflex put dup 230 /Ecircumflex put dup 231 /Aacute put dup 236 /Icircumflex put dup 237 /Iacute put dup 238 /Edieresis put dup 239 /Idieresis put dup 253 /yacute put dup 254 /Thorn put pop /re_dict 4 dict def /ref { re_dict begin /newfontname exch def /basefontname exch def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict { exch dup dup /FID ne exch /Encoding ne and { exch newfont 3 1 roll put } { pop pop } ifelse } forall newfont /FontName newfontname put newfont /Encoding mt_vec put newfontname newfont definefont pop end } def /Times-Roman /MT_Times-Roman ref 384 /MT_Times-Roman f1 (\303) 73 565 sh (\303) 2133 866 sh /f2 {ff matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (a) -30 611 sh (s) 2029 912 sh 224 /MT_Times-Roman f1 (4) 244 707 sh (1) 1363 984 sh (4) 2822 152 sh 384 ns (1) 861 362 sh 9384 /Symbol f1 (=) 569 611 sh (-) 2143 362 sh (\346) 1548 424 sh (\350) 1548 843 sh (\366) 2620 424 sh (\370) 2620 843 sh 224 ns (=) 1235 984 sh 576 ns (\345) 1099 698 sh 384 /Times-Italic f1 (n) 863 912 sh (x) 1788 362 sh (x) 2391 362 sh 224 ns (i) 1961 458 sh (i) 1147 984 sh (n) 1248 209 sh end MTsave restore dMATHQ "a  4 =1nx i -2x"s () i=1n   4FMicrosoft Equation Editor 2.0DNQE Equation.2 "a  4 =1nx i -2x"s () i=1n   4dr,ObjInfoSU Equation Native  _944648034XF00Ole  PIC WZdMETA PICT Y\) CompObj<Yr    .  & Times |w|wgw -! PSymbol |w|wgw  -!a Times |w|wgw -!4 & ' )dxpr  " currentpoint ",Times .+  , Symbol( a  +4/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 448 div 480 3 -1 roll exch div scale currentpoint translate 64 48 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def /mt_vec StandardEncoding 256 array copy def /Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute /agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave /ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde/oacute /ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex/udieresis mt_vec 128 32 getinterval astore pop mt_vec dup 176 /brokenbar put dup 180 /twosuperior put dup 181 /threesuperior put dup 188 /onequarter put dup 190 /threequarters put dup 192 /Agrave put dup 201 /onehalf put dup 204 /Igrave put pop /Egrave/Ograve/Oacute/Ocircumflex/Otilde/.notdef/Ydieresis/ydieresis /Ugrave/Uacute/Ucircumflex/.notdef/Yacute/thorn mt_vec 209 14 getinterval astore pop mt_vec dup 228 /Atilde put dup 229 /Acircumflex put dup 230 /Ecircumflex put dup 231 /Aacute put dup 236 /Icircumflex put dup 237 /Iacute put dup 238 /Edieresis put dup 239 /Idieresis put dup 253 /yacute put dup 254 /Thorn put pop /re_dict 4 dict def /ref { re_dict begin /newfontname exch def /basefontname exch def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict { exch dup dup /FID ne exch /Encoding ne and { exch newfont 3 1 roll put } { pop pop } ifelse } forall newfont /FontName newfontname put newfont /Encoding mt_vec put newfontname newfont definefont pop end } def /Times-Roman /MT_Times-Roman ref 384 /MT_Times-Roman f1 (\303) 73 258 sh /f2 {ff matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (a) -30 304 sh 224 /MT_Times-Roman f1 (4) 244 400 sh end MTsave restore d'MATH "a  4 dFMicrosoft Equation Editor 2.0DNQE Equation.2 "a  4ObjInfo[]>Equation Native ?7_944648261Vv`Fs0s0Ole @dr@r    .  & Times |w|wgw -! PSymbol[ b|w|wgw[ b -!a TimesPIC _bAdMETA CPICT adJ| CompObjpY |w|wgw -!3 & ' | dxpr  " currentpoint ",Times .+  , Symbol~@( a  @+3/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 448 div 512 3 -1 roll exch div scale currentpoint translate 64 48 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def /mt_vec StandardEncoding 256 array copy def /Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute /agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave /ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde/oacute /ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex/udieresis mt_vec 128 32 getinterval astore pop mt_vec dup 176 /brokenbar put dup 180 /twosuperior put dup 181 /threesuperior put dup 188 /onequarter put dup 190 /threequarters put dup 192 /Agrave put dup 201 /onehalf put dup 204 /Igrave put pop /Egrave/Ograve/Oacute/Ocircumflex/Otilde/.notdef/Ydieresis/ydieresis /Ugrave/Uacute/Ucircumflex/.notdef/Yacute/thorn mt_vec 209 14 getinterval astore pop mt_vec dup 228 /Atilde put dup 229 /Acircumflex put dup 230 /Ecircumflex put dup 231 /Aacute put dup 236 /Icircumflex put dup 237 /Iacute put dup 238 /Edieresis put dup 239 /Idieresis put dup 253 /yacute put dup 254 /Thorn put pop /re_dict 4 dict def /ref { re_dict begin /newfontname exch def /basefontname exch def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict { exch dup dup /FID ne exch /Encoding ne and { exch newfont 3 1 roll put } { pop pop } ifelse } forall newfont /FontName newfontname put newfont /Encoding mt_vec put newfontname newfont definefont pop end } def /Times-Roman /MT_Times-Roman ref 384 /MT_Times-Roman f1 (\303) 73 258 sh /f2 {ff matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (a) -30 304 sh 224 /MT_Times-Roman f1 (3) 237 400 sh end MTsave restore d'MATH "a  3 dFMicrosoft Equation Editor 2.0DNQE Equation.2 "a  3dr,r    .ObjInfocerEquation Native s7_944648273hFp0p0Ole tPIC gjudMETA wPICT il~| CompObjY  & Times |w|wgw -! PSymbol y|w|wgw y -!a Times |w|wgw -!4 & ' | dxpr  " currentpoint ",Times .+  , Symbol~@( a  @+4/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 448 div 480 3 -1 roll exch div scale currentpoint translate 64 48 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def /mt_vec StandardEncoding 256 array copy def /Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute /agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave /ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde/oacute /ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex/udieresis mt_vec 128 32 getinterval astore pop mt_vec dup 176 /brokenbar put dup 180 /twosuperior put dup 181 /threesuperior put dup 188 /onequarter put dup 190 /threequarters put dup 192 /Agrave put dup 201 /onehalf put dup 204 /Igrave put pop /Egrave/Ograve/Oacute/Ocircumflex/Otilde/.notdef/Ydieresis/ydieresis /Ugrave/Uacute/Ucircumflex/.notdef/Yacute/thorn mt_vec 209 14 getinterval astore pop mt_vec dup 228 /Atilde put dup 229 /Acircumflex put dup 230 /Ecircumflex put dup 231 /Aacute put dup 236 /Icircumflex put dup 237 /Iacute put dup 238 /Edieresis put dup 239 /Idieresis put dup 253 /yacute put dup 254 /Thorn put pop /re_dict 4 dict def /ref { re_dict begin /newfontname exch def /basefontname exch def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict { exch dup dup /FID ne exch /Encoding ne and { exch newfont 3 1 roll put } { pop pop } ifelse } forall newfont /FontName newfontname put newfont /Encoding mt_vec put newfontname newfont definefont pop end } def /Times-Roman /MT_Times-Roman ref 384 /MT_Times-Roman f1 (\303) 73 258 sh /f2 {ff matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (a) -30 304 sh 224 /MT_Times-Roman f1 (4) 244 400 sh end MTsave restore d'MATH "a  4 dFMicrosoft Equation Editor 2.0DNQE Equation.2 "a  4ObjInfokmEquation Native 7_944648394pFP0P0Ole d d X     .  & Times |w|wgw -!JBPSymbol |w|wgw -!=TimesPIC ordMETA PICT qt CompObjY |w|wgw -!n Times |w|wgw -!6-! ,PSymbol |w|wgw -!a (Times |w|wgw -!31PSymbol |w|wgw -!(%!)5Times |w|wgw  -!2 9PSymbol |w|wgw -!+ATimes |w|wgw  -!1 J!4JIP! 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