The title of this section really just says “applied statistics for financial analysis.”  Some of the basic principles in the quant section will appear in other exam sections, and this series will attempt to highlight such items.

There will likely be one item set (six questions), but two is a possibility.  Try not to let statistics un-necessarily weigh on your test pass strategy.  Depending on your background, you may have seen some or all of this material; former students of undergrad economics or MBA programs should have some prior exposure.

Material

The material covered in this section includes:
<ol>
	<li>Correlation and Single Variable Regression Analysis</li>
	<li>Multiple Variable Regression Analysis</li>
	<li>Time Series Analysis</li>
</ol>
The profession of financial analyst requires a basic ability to create theories about relationships and testing those theories.  For example, can one or more economic indicators predict the next twelve months return generated by a stock market index?  An analyst may seek to find a statistically valid relationship ahead of his peers, and exploit this knowledge in an attempt to generate “excess returns” (as that concept is defined in a given investment situation).

Tips:
<ul>
	<li style="margin-bottom: 10px;">If you have never seen a model output report from some commercial stats software, then you will need to take time to get comfortable with the format of cross sectional regression and time series model reports.  Your item set questions may center on successfully interpreting an example stats report.</li>
	<li style="margin-bottom: 10px;">The general guidance is to focus first on understanding the applied concepts (interpreting model outputs and identifying potential model pitfalls) and then incorporate the math and try to memorize some key formulas.</li>
	<li style="margin-bottom: 10px;">Like every exam section, practicing problems is a must.  If you are in the conceptual learning stage for quant, then you can read and try to work conceptual/interpretive questions in practice item sets, and return later to those same item sets to test your hand at the “pure math.”</li>
	<li style="margin-bottom: 10px;">Candidates with limited stats experience can learn enough to pick up two or three points in a six point item set.</li>
</ul>

CFA L2: Quantitative Methods - Introduction

Correlation

<ul>
	<li>Correlation is math-speak for relationships.  Is there a relationship between the change in the value of one variable and the change in value of another?</li>

	<li>Correlation and simple regression can help you:</li>
<ol>
<li>Verify a relationship between dependent variable Y and independent variable X.</li>

<li>Identify the mathematical form of the relationship (ex. linear, exponential)</li>

<li>Determine the value of the y-intercept and the slope of the coefficient.</li>
</ol>
</ul>

Correlation Coefficient

<ul>
	<li>Correlation Coefficient = a range between -1 and 1</li>
<ul>
	<li>Determines the direction (positive or negative) and strength of the relationship (a value of zero indicates no relationship) between two variables.</li>
	<li>Commonly expressed as “ryx”</li>
	<li>This value must be tested for significance in order to determine if developing a single regression model is merited.</li>
	<li>The null (Ho) hypothesis assumes that ryx = 0 and no relationship exists.</li>
	<li>Look at diagrams for a Student’s t-distribution to visualize your null hypothesis’ fail to reject and rejection ranges.</li>
	<li>If you believe that you have found a relationship, then your hope is that the null will be rejected and the correlation coefficient is not equal to zero.</li>
</ul>
</ul>

Limits of Correlation Analysis

<ul>
	<li>The correlation coefficient assumes that the relationship is liner, but many relationships between two variables are non-linear.</li>
	<li>If the data sample contains outlier observations, then rxy can be distorted.</li>
	<li>The analyst may discover a high correlation when no real relationship exists (the relationship is spurious).</li>
<ul>
	<li>Data mining for relationships is not preferred to holding an actual theoretical basis for testing to identify potentially significant correlations.</li>
</ul>
</ul>

Quants: Correlation Analysis

<ul>
	<li>This is the “least squares” method.</li>
</ul>
<ul>
	<li>Situational Example:</li>
</ul>
<ul>
<ul>
	<li>“If a country’s broad stock index (X) appreciates 5%, by how much will the value of that index’s largest air transportation stock change?”</li>
	<li>You will apply this type of modeling in the equity section, when looking at the CAPM approach to asset valuation.</li>
</ul>
</ul>
<ul>
	<li>Simple Regression can be expressed as follows:
<div class="formula1">Yi = b0 + b1Xi + ei
<ul>
	<li>b0 = the y-intercept.  Yi = b0, when observation Xi = 0 (zero)</li>
	<li>b1 = the slope coefficient – the change in Y per unit change in X.</li>
	<li>Xi = the observation for the independent variable.  Ex: the value of a stock index.</li>
	<li>Yi = the observed dependent variable for Xi.</li>
	<li>ei = the error term or the part of the dependent value not explained by the independent variable; the expected value of the error term is zero and this is one of the standard assumptions for simple regression.</li>
	<li>Note: the y-intercept and slope coefficient are the known as the model’s parameters.</li>
</ul>
</div></li>
</ul>
<ul>
	<li>A hat “^” is typically used when referring to predicted values and the subscript “i” typically refers to actual observations.</li>
</ul>
<h3>Six Assumptions of Simple Linear Regression</h3>
<ol>
	<li>Y and X must have a liner relationship.</li>
	<li>X is not random.</li>
	<li>The expected value of e is 0 (zero).</li>
	<li>The e term does not exhibit heteroskedasticity, meaning that the error term’s variance is the same for all observations.</li>
	<li>The error term is uncorrelated across all observations (or no serial correlation).</li>
	<li>The error term has a normal distribution.</li>
</ol>
A violation of one or more of these assumptions threatens the validity of your model’s conclusions.

Quants: Single Variable Linear Regression Analysis

<ul>
	<li>Also called Standard Error of the Regression</li>
	<li>Conceptually SEE helps to measure how imperfect your model is at predicting the value for a dependent variable, Y.</li>
	<li>Mathematically SEE can be seen a measure of deviation(s) around your model’s regression line, a line one deviation above and below the predicted equation line</li>
</ul>


The following video explains the concept using sample data from Google and yahoo stock quotes.

<iframe width="600" height="437" src="http://www.youtube.com/embed/dJR1WqeBgCg" frameborder="0" allowfullscreen></iframe>

Standard Error of the Estimate or SEE

<ul>
	<li>In all likelihood, your model will not perfectly predict Y.</li>
	<li>The <a href="https://financetrain.com/standard-error-of-the-estimate-or-see/" title="Standard Error of the Estimate or SEE">SEE</a> can be extended to determine the confidence interval for a predicted Y value.  A common CI to test for a predicted value is 95%.</li>
	<li>Your <a href="https://financetrain.com/quants-single-variable-linear-regression-analysis/" title="Quants: Single Variable Linear Regression Analysis">regression parameters</a>, the y-intercept (b0) and slope coefficient (b1) will need to be tested for significance before you can generate a confidence interval around your model’s project Y value around an expected X value.</li>
<ul>
	<li>H0 = 0 is the null hypothesis when testing either parameter and you will look to reject this in significance, (note: typically the greater emphasis is on the slope coefficient, as b1 value not statistically different from zero indicates no relationship between Y and X).</li>
	<li>tcalc = the standard script for the output of your significance test on the regression model’s parameters and its absolute value must exceed the designated tcritical on a two tailed significance test.</li>
</ul>
</ul>

Confidence Intervals (CI) for Dependent Variable Prediction

<ul>
	<li>Typically noted as R2yx or R-squared in the stats report.</li>
	<li>This value measures the percentage of variation in Y that is explained by the model and will be between 0 and 1 (and not to be confused with the Correlation Coefficient which will be between -1 and 1).</li>
	<li>Example: a coefficient of determination/R-squared = .80 would mean that 80% of the variation in dependent Y variable is explained by the model’s regression equation.</li>
	<li>For single variable/simple regression, the coefficient of determination equals the square of the data sample’s correlation coefficient.</li>
</ul>

The following video from Khan Academy explains the calculation of the Coefficient of Determination.

<iframe width="600" height="335" src="http://www.youtube.com/embed/lng4ZgConCM" frameborder="0" allowfullscreen></iframe>

Coefficient of Determination (R-Squared)

This is a tool to review a regression analysis and decompose the contribution of the variation in the independent X variable and the variation in the error (or residual) term in predicting the variation in the dependent Y variable.

Regression Sum of Squares (RSS) measures the amount of variation in Y that is explained by the variation in X.
<div class="formula1">Mean Sum of Squares of the Regression (MSSR) = RSS/degrees of freedom</div>
Sum of the Squares of the Errors (SSE) measures the amount of variation in Y that is explained by the variation in the error term.
<ul>
	<li>Mean Sum of Squares of the Errors (MSSE) = SSE/(n - x variables - 1)
<div class="formula1">√MSSE = SEE</div></li>
	<li>Be careful not to confuse SSE with the Standard Error of the Estimate (SEE) – SEE is one standard deviation above and below the model line, while SSE is measure of the error term’s contribution to the explanation of the dependent variable.</li>
</ul>
Total Sum of Squares (TSS) indicates the total amount in variation in Y and equals RSS + SSE.

Note that mathematical relationships exist among R2yx, RSS, SSE, and TSS, enabling the analyst to derive one value, if given values for the others.  Knowing these relationships can save you time on the exam and greatly simplify a problem that appears complex.

Analysis of Variance or ANOVA

Much of the concepts in simple regression are applicable, but watch out when determining your degrees of freedom for different analyses, as the values will be slightly different for models similar in observation count, but different in slope coefficient count.</li>

<h3>Six Assumptions of Multiple Regression (very similar to simple regression)</h3>
<ol>
<li>Y and X must have a liner relationship.</li>

<li>X is not random and there is no multicollinearity among two or more of the independent variables.</li>

<li>The expected value of e is 0 (zero).</li>

<li>No heteroskedasticity, i.e., the error term’s variance is the same for all observations /does not exhibit a relationship with the independent variable.</li>

<li>No serial correlation, i.e. error terms are uncorrelated with one another across all observations.

<li>The error term has a normal distribution.</li>

</ol>

Be sure to review and get comfortable with the standard form of a statistical software program’s output for a multiple variable regression analysis.
<ul>
	<li>Example multiple regression equation: Yi = b0 + b1X1i + b2X2i  + ei</li>
	<li>You will need to be able to:</li>
<ul>
	<li>Use this equation to estimate a dependent variable (this becomes simple plug and chug once you are comfortable with the material)</li>
	<li>Test the overall validity of a multiple regression model (see Fcalc below)</li>
	<li>Perform tcalcs to test the validity of the y-intercept and individual slope coefficients.</li>
	<li>Determine confidence intervals for individual slope coefficients.</li>
	<li>Hypothesis testing to determine if a slope coefficient is statistically different from some specified value (maybe your colleague creating a similar model but deriving a different slope coefficient; this testing will determine if the difference is statistically significant).</li>
	<li>Determine the Standard Error of the Estimate (SEE)</li>
</ul>

Multiple Regression Analysis

<ul>
	<li>For a multiple regression model, this value represents the percentage of total variation in Y that is explained by the regression equation.</li>
	<li>The value is between 0 and 1.</li>
	<li>R-squared has a mathematical relationship with TSS, SSE, and RSS.</li>
	<li>R2 = RSS/TSS = (TSS-SSE)/TSS = 1- (SSE/TSS)</li>
	<li>The coefficient of determination alone does not indicate that a model is well specified, for example you could have more independent variables than necessary and the R2 will still be high – in this case your model would be not be considered parsimonious.</li>
	<li>Adjusted R2 = an alternate measure and will always be smaller than R2</li>
</ul>

Multiple Regression and Coefficient of Determination (R-Squared) 

<ul>
	<li>A statistically significant Fcalc (i.e. one that passes the Fcritical threshold, based on your degrees of freedom) can indicate that your model as a whole is meaningful.</li>
	<li>This test is really applicable for <a title="Multiple Regression Analysis" href="https://financetrain.com/multiple-regression-analysis/">multiple regressions</a>, where there is more than one slope coefficient (b1, b2, b3 … bi), as a t-test will not work for multiple regression models.</li>
	<li>The F-test is a one tailed test.</li>
	<li>The null hypothesis will be that the Fcalc is less than or equal to the Fcritical and you will be looking to reject the null with an Fcalc &gt; Fcritical.</li>
	<li>A rejection of the null indicates that at least one of the slope coefficients is significant and there is some validity to the model.</li>
	<li>Fcalc has a math relationship with RSS (and MSSR), <a title="Standard Error of the Estimate or SEE" href="https://financetrain.com/standard-error-of-the-estimate-or-see/">SSE</a> (and MSSE), and TSS.</li>
</ul>

Fcalc – the Global Test for Regression Significance

Heteroskedasticity

There are two types, Conditional and Unconditional.  The type focused on in evaluating model validity is Conditional Heteroskedasticity.
<ul>
	<li>Conditional = the error terms change in a systematic manner that is correlated with the values of the independent variables.</li>
<ul>
	<li>Look up a graph depicting this problem.</li>
</ul>
	<li>The Breusch-Pagan test will test for Conditional Heteroskedasticity.</li>
	<li>When this problem is present, the model’s t-scores will be artificially high, indicating a false significance of relationships.</li>
</ul>
Serial Correlation

This is interaction of your model’s error terms.
<ul>
	<li>When <a title="What is Serial Correlation (Autocorrelation)?" href="https://financetrain.com/serial-correlation-autocorrelation/">serial (or auto) correlation</a> is present your SEE may be incorrect.</li>
	<li>The Durbin-Watson test statistic can be used to determine the presence of Serial Correlation in multiple regression models, as well as simple and log linear time series models, but not on auto-regressive time series models.</li>
</ul>
Multi-collinearity 

Two or more of your independent variables are highly correlated.
<ul>
	<li>A tiny bit of multi-collinearity is tolerable and can be common in regression models involving several independent variables.</li>
	<li>A common symptom of this problem is the presence of a high <a title="Multiple Regression and Coefficient of Determination (R-Squared)" href="https://financetrain.com/multiple-regression-and-coefficient-of-determination-r-squared/">coefficient of determination (R2)</a>, despite having low t-scores for your independent variables (i.e. they are insignificant).</li>
</ul>

Regression Analysis and Assumption Violations

<ul>
	<li>Handle qualitative independent variables with a quantitative proxy or use a dummy variable.</li>
	<li>When using a dummy independent variables (such as assigning a number to the degree of consumer confidence), define a collectively exhaustive set of “j” categories, then j-1 (“j minus one”) will give you the number of dummy variables for inclusion in your model.</li>
	<li>Models with dummy independents can easily be misspecified.</li>
</ul>
Model types with qualitative dependent variables
<ul>
	<li>Probit models – based on a normal distribution and attempt to estimate the probability that the dependent variable will equal 1.</li>
	<li>Logit models – based on the logistic distribution and like Probit models, they attempt to estimate the probability that the dependent variable will equal 1.</li>
	<li>Discriminant Analysis – creates a score and if the score crosses a threshold then the dependent variable is assigned a 1.</li>
</ul>
Looking at the big picture, you want your <a title="Multiple Regression Analysis" href="https://financetrain.com/multiple-regression-analysis/">multiple regression model</a> to:
<ol>
	<li>Have a good theoretical basis and;</li>
	<li>Pass the most stringent statistical tests (refer back to the sub-section “<a title="Regression Analysis and Assumption Violations" href="https://financetrain.com/regression-analysis-and-assumption-violations/">Assumption Violations</a>”).</li>
</ol>

Qualitative and Dummy Variables in Regression Modeling

Simple Time Series Models

<ul>
	<li>This is basic trend modeling.</li>
</ul>
<div class="formula1">A simple trend model can be expressed as follows:

yt = b0 + b1t+ &epsilon;t

<ul>
	<li>b0 = the y-intercept; where t = 0.</li>
	<li>b1 = the slope coefficient of the time trend.</li>
	<li>t = the time period.</li>
	<li>ŷt = the estimated value for time t based on the model.</li>
	<li>ei = the random error of the time trend.</li>
</ul>
</div>
<ul>

	<li>The big validity pit-fall for simple trend models is serial correlation; if this problem is present, then you will see an artificially high R2 and your slope coefficient may falsely appear to be significant.</li>
	<li>There is a visual way to detect serial correlation (not shown) or you can perform a Dubin-Watson test.</li>
</ul>
Log-linear Trend Models
<ul>
	<li>This applies to non-linear time series trends.</li>
</ul>

<div class="formula1">
The structure is:

<ul>
	<li>ln yt = b0 + b1t+ et; or</li>
	<li>yt = e b0 + b1t + et</li>
</ul>
</div>
<ul>

	<li>Again, like the simple trend model, use a graph or Durbin Watson test to check for serial correlation, as this will be a big threat to validity.</li>
</ul>

Time Series Analysis: Simple and Log-linear Trend Models

<ul>
	<li>Auto-Regressive (AR) Time Series Models</li>
	<li>This type of time series model utilizes a time period lagged observation as the independent variable to predict the dependent variable, which is the value in the next time period.</li>
</ul>

<div class="formula1">
	xt = b0 + b1xt-1 + &epsilon;t
</div>

<ul>
	<li>There can be more than one time period lag independent variable.</li>
	<li>Valid statistical inferences from AR time series models only if the time series is covariance stationary; a time series with growth over time or seasonality is not covariance stationary.</li>
	<li>It is critical to test your AR time series model for serial correlation and the Durbin-Watson test cannot be used for this model.</li>
	<li>An AR time series model that is covariance stationary will exhibit mean reversion – it will tend to fall after going above the mean and rise after going below the mean.</li>
	<li>Root Mean Square Error (RMSE) = a method of assessing the out of sample accuracy of a time series model’s forecast.  If comparing multiple models, the model will the lowest RMSE is considered to have the best forecasting capabilities.</li>
</ul>

Auto-Regressive (AR) Time Series Models

<ul>
	<li>This is the case of an AR time series model where the predicted value is expected to equal the previous period plus a random error: </li>
<div class="formula1">
xt = b0 + xt-1 + &epsilon;t
</div>
	<li>When b0 is not equal to zero, the model is a random walk with a drift, but the key characteristic is a b1 = 1.</li>
	<li>The expected value of the error is still zero.</li>
	<li>The mean reverting level for a random walk is not covariance stationary and the technique of first differencing is frequently used to transform an AR model with one time lag variable (AR1) into a model that is covariance stationary.</li>
	<li>If an AR time series is covariance stationary, then the serial correlations for the lag variables are insignificant or they rapidly drop to zero as the number of time period lags rises.</li>
	<li>When the lag coefficient is not statistically different from 1, a unit root exists.</li>
	<li>Dickey-Fuller test = applied to AR1 model to test for a unit root.</li>
	<li>If a unit root is present, then the model is not covariance stationary; if this is the case, the independent variable must be transformed, so you can re-model.</li>
</ul>

Auto-Regressive Models - Random Walks and Unit Roots

<ul>
	<li>Autoregressive Moving Average Model (ARMA) = calculates an average value over a period of time to smooth fluctuations in a time series.</li>
	<li>ARMA models are very sensitive to minor changes and may rarely forecast well.</li>
	<li>Auto Regressive Conditional Heteroskedasticity (ARCH) testing = can be used to determine if an AR, MA, or ARMA model suffers from conditional heteroskedasticity.</li>
	<li>The ARCH test models the error terms and if its slope is statistically significant, then the predictive AR, MA, or ARMA model under scrutiny is not valid.</li>
</ul>

ARMA Models and ARCH Testing

<ul>
	<li>Simple Linear and Exponential Growth Models – If an analyst looks at a time series plot graph he/she may see patterns exhibiting possible linear or exponential growth relationship to the dependent variable.  Serial correlation of the error terms must not be present and the Durbin Watson test can test for this.</li>
	<li>Auto-Regressive Models – If serial correlation exists in a simple time series model, the analyst can create an auto-regressive time series with the sample data, where the independent variable is a lagged (prior period) value.  The AR model is appropriate where the prior period value is the best predictor for the future period dependent variable value.  The D-W test will not work on an AR model, so the analyst needs to examine the error term correlations to check for the presence of serial correlation.  A Dickey Fuller test can test for the presence of a unit root in the AR model.</li>
	<li>Seasonality –The plot of a time series model may show seasonality; the model may be improved by adding a seasonal lag variable, through the technique of first differencing.</li>
	<li>Moving Average and AR Moving Average Models – This model modification may improve on a base AR model.</li>
	<li>Auto Regressive Conditional Heteroskedasticity – ARCH must be tested for to ensure that the AR, MA, or ARMA model’s t-scores are not overstated.</li>
	<li>Out of Sample Data Testing – If an analyst has several time series models examining a dependent variable, then the forecasted values for each can be compared with actual out of sample data values.  The model with the lowest root mean square error (RMSE) is considered to have the best predictive capabilities.</li>
</ul>

How to Select the Most Appropriate Time Series Model?

Quantitative Methods

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