Computer Exercises for Econometrics

Transcription

1 Computer Exercises for Econometrics Simon Broda August 30,

2 1 Preliminaries Prerequisites Know how to turn on a computer Topics Covered MATLAB L A TEX(the program this great manual was produced with) Scientific Workplace / Maple (possibly) Required Reading The Matlab Primer. Available at and various other places on the web. The first one or two sessions will be based on this. Handouts I will give you. The Not So Short Introduction to L A TEX2e. Available in many different languages at User s Guide for the amsmath Package. Available at dunbar/docs/amsldoc.pdf 2 MATLAB Basics When you start MATLAB, you will be presented a window with 3 parts: 2

3 The right-hand part, the so-called command window, is the most important one: this is where you will enter commands to tell MATLAB what you want it to do. The left half of the screen contains a list of your variables and a command history, but we will not need them now. For the beginning, you may think of MATLAB as nothing but a calculator. Try it: In the command window, enter something like 5+5, followed by the <ENTER> key. The commands for the other basic arithmetic operations are -, / (division), * (multiplication) and ^ ( to the power of ). The single most important MATLAB command is help <functionname>. When invoked without an argument, help will display a list of available help topics: >> help HELP topics: matlab\general - General purpose commands. 3

4 matlab\ops - Operators and special characters. matlab\lang - Programming language constructs. matlab\elmat - Elementary matrices and matrix manipulation. matlab\elfun - Elementary math functions. matlab\specfun - Specialized math functions. matlab\matfun - Matrix functions - numerical linear algebra. matlab\datafun - Data analysis and Fourier transforms. matlab\audio - Audio support. matlab\polyfun - Interpolation and polynomials. matlab\funfun - Function functions and ODE solvers. matlab\sparfun - Sparse matrices. matlab\graph2d - Two dimensional graphs. matlab\graph3d - Three dimensional graphs. matlab\specgraph - Specialized graphs. matlab\graphics - Handle Graphics. matlab\uitools - Graphical user interface tools. matlab\strfun - Character strings. matlab\iofun - File input/output. matlab\timefun - Time and dates. matlab\datatypes - Data types and structures. matlab\verctrl - Version control. matlab\winfun - Windows Operating System Interface Files (DDE/COM) winfun\comcli - (No table of contents file) matlab\demos - Examples and demonstrations. toolbox\local - Preferences. toolbox\exlink - Excel Link. finance\finance - Financial Toolbox. finance\calendar - Financial Toolbox calendar functions. finance\findemos - Financial Toolbox demonstration functions. finance\finsupport - (No table of contents file) toolbox\optim - Optimization Toolbox toolbox\stats - Statistics Toolbox MATLAB6p5\work - (No table of contents file) For more help on directory/topic, type "help topic". For command syntax information, type "help syntax". 4

5 Note that MATLAB s functions are organized in so-called toolboxes. For example, we will often need the statistics toolbox, help on which is accessed by entering help stats. Entering help <topic> will display help on a specific topic. >> help sin SIN Sine. SIN(X) is the sine of the elements of X. Clearly, this is of little help when the name of the MATLAB function is not known. In this case, the command lookfor <searchphrase> is useful; it returns a list of MATLAB commands the help texts of which contain the searchphrase: >> lookfor sine ACOS Inverse cosine. ACOSH Inverse hyperbolic cosine. ASIN Inverse sine. ASINH Inverse hyperbolic sine. COS Cosine. COSH Hyperbolic cosine. SIN Sine. SINH Hyperbolic sine. TFFUNC time and frequency domain versions of a cosine modulated Gaussian pulse. xfourier.m: %% Square Wave from Sine Waves BUSDATE Next or previous business day. FBUSDATE First business date of month. ISBUSDAY True for dates that are business days. LBUSDATE Last business date of month. Scalar Functions The sin command mentioned earlier is one example of a MATLAB function. A function call is a special command which consists of the function name and, in parentheses, the function argument: 5

6 >> sin(5) ans = Other such elementary functions are exp (exponential), log (natural logarithm), sqrt (square root), abs (absolute value), etc; type help elmat for a comprehensive list. Some functions require more than one input argument. For example, normrand(<mu>,<sigma>) will return a random number from the normal distribution with mean mu and scale sigma. Matrices and Vectors Arguably the greatest strength of MATLAB is its ability to handle vectors and matrices; in fact, the name MATLAB stands for MATrix LABoratory. A matrix is entered by listing its individual elements in square brackets, separating lines with semicolons (or linefeeds) and columns with spaces: >> [ ; ] ans = Most of the elementary functions above will work for matrices; they will then operate elementwise: >> sin([ ; ]) ans = In order to make the basic arithmetic functions *, /, and ^ operate elementwise, the respective operator must be preceeded by a dot: 6

7 >> [ ].*[ ] ans = Of course, as a Matrix Laboratory, MATLAB also features special functions for matrices, some of which are (transpose), inv (inverse), eig (eigenvalues), chol (cholesky decomposition), etc; type help elmat for a list. Also, function exist for the construction of special matrices. For example, eye(<n>) will return a n n identity matrix. Similarly, ones(<n>,<m>) will produce a n m matrix of ones, while zeros(<n>,<m>) will produce a n m matrix of zeros. Variables Of course, your shiny new calculator also has the ability to store results. MATLAB uses variables for this purpose. A variable is nothing more than some reserved space in memory with a name to it. To assign a value to a variable, use the symbol = : A=[ ; ; ] A = Variable names can be made up from letters, numbers and the underscore ( ). They may not, however, start with a number. Note that MATLAB is case-sensitive: A is not the same as a. If A is a matrix, in order to access the (i,j)th element of A, write A(i,j): A=[ ; ; ] A = 7

8 >> A(2,2) ans = 6 To access a row (or column) of A as a whole, write A(i,1:end) or simply A(i,:). The special variable ans always contains the most recent result. The commands who and whos display information on current variables. The information returned by whos is also displayed in the upper left half of the MATLAB window. Variables can be saved to disk for later use with the command save <filename> <variable names>. If <variable names> is omitted, all variables will be saved. The command load <filename> <variable names> works analogously, but loads variables from file <filename>. Relational Operators Relational operators<,<=,>,>=,== (is equal), and = (is not equal) serve the purpose of comparing variables and will return 1 if the relation is true and 0 otherwise. Note the difference between == and = which is used in assignment statements: >> a=5 a = 5 >> a==5 ans = 8

9 1 Relations can be connected with the logical operators & (and), (or), and (not). For example, the following statement will always return 1 unless a = 1: (a<1) (a>1). m-files Until now, we have told MATLAB what we wanted it to do by just typing commands into the command window. This is clearly impractical if we want to reproduce the results later. Instead, it is possible put the commands into a file, called an m-file. m-files can be comfortably edited in the MATLAB editor. On a Windows machine, the MATLAB editor can be invoked from the start menu, or, usually more conveniently, by typing edit <filename> in the MATLAB command window. If <filename> does not exist, the editor will ask you if you want to create one upon starting up. m-files can be executed by simply typing their name in the command window, very much like an ordinary MATLAB function. You may want to try the following example program. In the command windows, type edit example1. The editor will ask you if you want to create a file by this name. Confirm by clicking Yes. Inside the editor window, type the following: randn( state,0) T=25,U=randn(T,1); X=[ones(T,1) [1:T] ];%the regressor matrix beta=[2;1];y=x*beta+u; betahat=inv(x *X)*X *Y Save the file to disk. From the command window, invoke our little example script file by typing example1: 9

10 >> example1 T = 25 betahat = So what can we learn from this example? The function randn(<n>,<m>) generates a n m matrix of iid standard normal (pseudo) random numbers. randn( state,0) resets the random number generator to its initial state so that the results can be reproduced. The % character starts a comment. Everything on the same line after that symbol will be ignored. You can have more than one command per line. They must be seperated by commas or semicolons; the latter will supress the output of the respective command. The command 1:T produces a row vector of the numbers 1... T in increasing order. Similarly, the command 1:2:2*T produces a vector of odd numbers ranging from 1 to 2T, i.e., the middle number determines the step size. Matrix definitions can be nested. The if-else-end Statement Sometimes we want to execute a certain command only if some condition is met This is where the if-else-end statement comes in. Its syntax is if <expression> <first command block> else <second command block> end 10

11 <expression> is an expression that produces 0 or 1 as a result. If <expression> is true, <first command block> will be executed, and <second command block> otherwise. For example, the statements if a<1 disp( a<1 ), else disp( a>1 );end will tell you if or not a > 1. It is also possible to have multiple if-branches; the syntax for this is: if <expression 1> <first command block> elseif <expression 2> <second command block> else <third command block> end Loops Sometimes we want to MATLAB to do something as long as a certain condition is met. This is achieved via the while statement. Its syntax is while <expression> end <command block> For example, the statements a=clock; while floor(a(4))<=18 & floor(a(4))>8 disp( You should be working. ); a=clock; end remind us of our working hours. Sometimes we want a loop to be executed a prespecified number of times. This could be achieved via a=1; while a<=100 11

12 end <commands> a=a+1; Since this kind of loop appears frequently, there is a shorthand notation for it: for a=1:100 end <commands> For example, the statements squares=zeros(25,1); for i=1:25; squares(i)=i^2; end; squares will store the squares of the numbers in a variable squares and display it. (Clearly, the shorter squares=[1:25].^2 would have achieved the same.) Functions So far, we have used m-files only as scripts, i.e., files that will be executed as if you had typed the commands in the command window. The other kind of m-file that MATLAB uses are user-defined function that can be used just like MATLAB s built-in functions. A function-file has a special syntax, as the following example illustrates: function result=sqr(x) %sqr(x) computes the square of the elements in x. result=x.^2; The above function must be saved in a file sqr.m 12

13 sqr is the name of the function, x is its input argument, and result is the return value. Note that variables used inside of a function are local, i.e., they have nothing to do with your workspace variables. To see this, try, in the command window: result=1;sqr(5);result and note that although the local variable result was assigned the value 5 inside the function, the workspace variable result still has the value 1. Clearly, the return variable must be assigned a value within the function, otherwise MATLAB will stop with an error message. The comment on the second line has a special purpose: It will be displayed by the help command. To see if it works, just type help sqr. Note that without the function declaration (the first line), sqr.m would be a script file. functions may have several output arguments, as the following example illustrates: function [themean,thevariance]=thestats(x) %thestats(x) computes the mean and variance of x. themean=sum(x)/length(x); thevariance=sum((x-themean).^2)/length(x); Once this is saved as thestats.m, the statement [a b]=thestats(randn(25,1)) will compute the mean and sample variance of a size 25 normal sample. Note that [a]=thestats(randn(25,1)) will only return the mean. Also, a function may have more than one input argument: function [themean,thevariance]=thestats2(x,unbiased) %thestats2(x) computes the mean and variance of x. themean=sum(x)/length(x); if unbiased==0 normalize=length(x); else normalize=length(x)-1; end thevariance=sum((x-themean).^2)/normalize; 13

14 thestats2 will return the MLE of the variance if unbiased==0 and the usual unbiased estimator otherwise. Now, of course, MATLAB will complain if thestats2 is called with only one argument. To avoid this, default values can be used: function [themean,thevariance]=thestats3(x,unbiased) %thestats3(x,unbiased) computes the mean and variance of x. if nargin<2 unbiased=1;end themean=sum(x)/length(x); if unbiased==0 normalize=length(x); else normalize=length(x)-1; end thevariance=sum((x-themean).^2)/normalize; the variable nargin contains the number of arguments the function was called with. Forcing Functions to Accept Vector Input For some purposes (e.g., plotting, quadrature; see below), it is necessary that a function accepts vector input. To understand what this means, recall function sqr.m from above: The way we implemented it, it does accept vector input. Had we instead programmed it like this function result=sqr(x) %sqr(x) computes the square of the elements in x. result=x^2;%note the missing dot before the ^ it would not. One way of forcing the function to accept vector input is the following: function result=sqr2(xvec) %sqr2(x) computes the square of the elements in x. result=zeros(size(xvec));%initialize to zero for loop=1:length(xvec)%work through the x vector x=xvec(loop); result(loop)=x^2; end; 14

15 Numerical Root Finding It often occurs that no closed-forms solution for the zeros of a function exists. The zeros must then be numerically obtained. For scalar functions, this could be achieved by, e.g., bisection or Newton s method, but it is more convenient to use the MATLAB function fzero. The syntax of the command is x=fzero(<fun>,<x0>,<options>,<p1>,<p2>,...). <fun> is the function we want to find the zeros of. It can be specified in several ways: As as string containing the name of the function. E.g., suppose we want to know a zero of the cosinus function. The syntax for this is x=fzero( cos,1) which will return π. 2 As a function handle: x=fzero(@cos,1) A function handle is a special data structure that contains all information about a function that MATLAB needs to evaluate it using feval, i.e., feval(@cos,1) is the same as cos(1). Type help function handle for more information. An inline object: x=fzero(inline( cos(x) ),1) Inline functions are a way of creating (simple) functions without having to make a function file. For example, the statement a=inline( randn(x,p1),1) will produce a function a that does the same as randn. Type help inline for more information. <x0> can be either a starting guess (when passed as a scalar) or an interval within which MATLAB will search for a zero (when passed as a 2-element vector). <options> is an object created by the optimset function. For example, optimset( TolX,1e-6) will set the tolerance on the position of the zero to To go with the default options, pass the word optimset. See help optimset for more details. fzero will search for a zero of the function with respect to its first argument. If the function takes more than one input argument, you may pass them as <p1>,<p2>,... 15

16 The following example computes the MLE for an iid Gam(a,b) sample. Remember that ˆb MLE = â MLE / X and â MLE is given implicitly as the solution to ln(a/ X) ψ(a)+ 1 n n i=1 ln x i = 0. function [a,b]=gammamle(x) a=fzero(@foc_a,1,optimset,x); b=a/mean(x); function result =foc_a(a,x) if a<=0 result=-a;return;end%quick and dirty result=log(a/mean(x))-psi(a)+mean(log(x)); To see if it works: >> a=3;b=5;randn( state,0);rand( state,0); >> X=gamrnd(a,1/b,100,1);[ahat bhat]=gammamle(x) ahat = bhat = (Note that MATLAB defines the Gamma distribution as Gamma(a,c), where c = 1/b, so that the second parameter is a genuine scaling parameter. For comparison, MATLAB s MLE routine gives: >> a=3;b=5;randn( state,0);rand( state,0); >> X=gamrnd(a,1/b,100,1);themle=mle( gam,x);[themle(1) 1/themle(2)] ans = For multidimensional root finding, fsolve must be used. 16

17 Numerical Optimization The MATLAB command for minimizing a scalar function of several variables is x=fminunc(<fun>,<x0>,<options>,<p1>,<p2>,...). fminunc uses a quasi-newton method (the BFGS algorithm). The input arguments correspond to those for the fzero command, except that for fminunc, <x0> must be starting value, the size of which must equal the number of variables over which to optimize. In our Gamma example, remember that the log likelihood is given by l(θ) = na ln b n ln Γ(a) + (a 1) n i=1 ln x i b n i=1 x i. The MLE could be implemented like this: function [a,b]=gammamle2(x) [ab]=fminsearch(@loglik,[1 1],optimset,X); a=ab(1);b=ab(2); function ll=loglik(ab,x) a=ab(1);b=ab(2); if a<=0 b<=0 ll=-a-b;return,end n=length(x); ll=-(n*a*log(b)-n*log(gamma(a))+(a-1)*sum(log(x))-b*sum(x)); Check if it works: >> a=3;b=5;randn( state,0);rand( state,0); >> X=gamrnd(a,1/b,100,1);[ahat bhat]=gammamle2(x) ahat = bhat =

18 Quadrature The term quadrature is used for numerical integration. The MATLAB command to accomplish this is quadl(<fun>,<a>,<b>,<tol>,<trace>,<p1>,<p2>,...), where <fun>, <p1>, <p2> are as above, [ab] is the interval over which to integrate, <TOL> is the absolute error tolerance, and <TRACE> will display interim results if set to anything other than zero. Use <TOL>=<TRACE>=[] to go with the defaults. Although <fun> must be a scalar function, it MUST accept vector input on which it operates elementwise; see above. The following example function will evaluate the normal cdf at <x>: function cdf=mynormcdf(x,mu,sigma) cdf=quadl(@normpdf,-1e+20,x,[],[],mu,sigma); Try what happens if -inf is used instead of -1e+20. For the evaluation of improper integrals of the form I = b a f(x) dx, with b =, the substitution t = 1/(x a + 1) proves useful, giving: Similarly, if a =, I = I = and if both a = and b =, I = 1 0 [ f f f ( a + 1 t ) 1 t t dt. 2 ( b 1 t ) 1 t t dt, 2 ( ) ( 1 t t 1 + f t t )] 1 t 2 dt. Application to our example involving the normal cdf yields function cdf=mynormcdf2(x,mu,sigma) cdf=quadl(@integrand,0,1,[],[],x,mu,sigma); function I=integrand(tvec,x,mu,sigma); 18

19 I=zeros(size(tvec)); for loop=1:length(tvec) t=tvec(loop); if (t==0 t==1) I(loop)=0; else u=x-(1-t)./t; I(loop)=normpdf(u,mu,sigma)./t.^2; end end Graphs and Printing To plot vector x versus vector y, use the MATLAB command plot(<x>,<y>). If the second argument is omitted, <x> will be plotted against its index. Example:x=[-3:0.01:3];plot(x,normpdf(x));. Various line types, plot symbols and colors may be obtained with plot(<x>,<y>,<s>) where s is a character string made from one element from any or all the following 3 columns: b blue. point - solid g green o circle : dotted r red x x-mark -. dashdot c cyan + plus dashed m magenta * star y yellow s square k black d diamond v triangle (down) ˆ triangle (up) < triangle (left) > triangle (right) p pentagram h hexagram For example, plot(<x>,<y>, c+: ) plots a cyan dotted line with a plus at each data point; plot(<x>,<y>, bd ) plots blue diamond at each data point but does not draw any line. 19

20 Use hold on to make a subsequent plots appear inside the same figure and hold off to have subsequent plot created in their own figures. a=figure creates an empty figure and returns its handle. figure(<a>) makes <a> the current figure. a=gcf gets the handle of the current figure; gca gets the handle of the current axis. grid toggles whether or not a grid is displayed on the current figure. axis([<xmin> <XMAX> <YMIN> <YMAX>]) sets the scaling for the x- and y-axes of the current figure. title( text ) adds a title to the current figure. Similarly, xlabel( text ) and ylabel( text ) set axis labels. set(<h>, PropertyName,PropertyValue) sets the value of the specified property for the graphics object with handle H. For example, set(gca, fontsize,16) will set the fontsize of the axis of the current figure to 16. set(<h>) will display a list of property names for the graphics object <H>, e.g., try set(gca). legend( string1, string2,..) will add a legend to the current figure. Try x=[-3:0.01:3];a=plot(x,normpdf(x));legend( Normal pdf ). print will send the current figure to the default printer. print -<driver> filename will use the specified driver and print to a file. E.g., use print -deps2 filename.eps for use with L A TEX. There are more options available for almost all commands listed above, but there is little use in covering every detail here. Please use the help facility. 20

21 3 L A TEX Obtaining L A TEX In the following, we will assume that you use MS Windows. If you use Linux (as you should), then chances are that all necessary programs are already installed or at least come bundled with your distribution. If you have a Macintosh, try cmactex ( From what I understand, the editor of choice for L A TEX on Macintosh is called Alpha and is available at Visit and download the file setup.exe. Run it. Choose Download only and enter a path to a temporary directory. After everything is downloaded, start setup.exe again, select Install and point it to your temporary directory. Visit Download and install the latest versions of GSview (4.5 at the time this was written) and Ghostscript (8.11 at the time this was written). Visit Download and install the latest version of WinEdt (5.3 at the time this was written). Note that unlike the other programs you just downloaded, WinEdt is NOT free. However, you may use it for 30 days without restrictions. Afterwards, WinEdt will start bugging you with requests to register it (unless you know how to avoid this ;-) Start WinEdt. Cancel the configuration wizard. Under Options Configurations, choose Miktex or Miktex Direct. Getting Started Unlike MS Word, L A TEX is not a WYSIWYG word processor, i.e., while editing a L A TEX document, you will not see right away what the result will look like. The document rather is a mixture of the actual text and specific commands that tell L A TEX what the final document should look like. Start WinEdt and create a new document. Save it as test.tex. Enter the following: 21

22 \documentclass{article} \begin{document} Hello World! \end{document} This is your first L A TEX document. To have it typeset, you have to compile the document by pressing the L A TEX button. WinEdt will call latex.exe, a DOS program which will produce the output file test.dvi. Afterwards, press the DVI button. This will open YAP, a DVI previewer. Unfortunately, the DVI format is not suitable for printing. If you want to print your document, you should convert it to a postscript file by clicking on the dvips button. Choose -P pdf -t A4 -z as generic parameters. To view the postscript file, click the button with the ghost with goggles to open GSview, the postscript viewer. To print it, select File Print... from within GSview. If you have a postscript printer (as is the case at ISB), it is recommended that you select PostScript Printer as print method. To convert the postscript document to a PDF file, press the ps2pdf button in WinEdt. There are other ways of creating PDF files: dvi2pdf will avoid the postscript step, and pdfl A TEX will directly compile to PDF, but these are NOT recommended. 22