InnerSoft STATS - Analyze

InnerSoft STATS
ANALYZE MENU
http://isstats.itspanish.org/

ANALYZE MENU InnerSoft STATS
INDEX
1. Frequency Tables
2. Descriptive Statistics
3. Crosstabs
4. One Sample Test
4.1. Z Test
4.2. T Test
4.3. Variance Test
5. Two-Sample Test
5.1. 2-Sample t-Test
5.2. Paired t-Test
5.3. 2 Variances F-Test
6. One-Way ANOVA
7. Homoscedasticity Tests
8. Bivariate Correlation Tests
9. Parametric Value at Risk
10. Exponentially Weighted Moving Average (EWMA) Forecast
11. Financial Formulas
12. Linear Regression
13. Curve Estimation
14. Create Time Series
15. Univariate GARCH(1,1)
2

1. – Frequency Tables
Overview
The frequency of a particular observation is the number of times the observation occurs in the data. The
distribution of a variable is the pattern of frequencies of the observation.
Frequency distribution tables can be used for both categorical and numeric variables. Use numeric codes
or strings to code categorical variables (nominal or ordinal level measurements).
Dialog box items
Order by. The frequency table can be arranged according to the actual values in the data or according to
the count (frequency of occurrence) of those values, and the table can be arranged in either ascending or
descending order.
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2. – Descriptive Statistics
Overview
Produces descriptive statistics for each column. The data columns must be numeric and contain at least
one numeric value. The dialog box allows to choose the statistics that you wish to display.
Available Variables list shows numeric data columns containing at least one no-missing value.
Dialog box items
Mean: Choose to display the arithmetic mean.
Sample Variance: Choose to display the unbiased variance of the data. Estimates population variance
based on a sample. If your data represents the entire population, then compute the variance by using Total
Variance.
Sample Standard Deviation: Choose to display the standard deviation of the data. Estimates population
standard deviation based on a sample. If your data represents the entire population, then compute the Std.
Deviation by using Total Std. Deviation.
Sample Coefficient of variation: Choose to display the coefficient of variation.
Sample Skewness: Choose to display the skewness value. Estimates population skewness based on a
sample. If your data represents the entire population, then compute the skewness by using Total
Skewness.
Sample Kurtosis: Choose to display the kurtosis value. Estimates population kurtosis based on a sample.
If your data represents the entire population, then compute the kurtosis by using Total Kurtosis.
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Total Variance: Choose to display the variance of the data. Calculates variance based on the entire
population.
Total Standard deviation: Choose to display the standard deviation of the data. Calculates Std. Deviation
based on the entire population.
Total Coefficient of variation: Choose to display the coefficient of variation. Calculates coefficient of
variation based on the entire population.
Total Skewness: Choose to display the skewness value. Calculates skewness based on the entire
population.
Total Kurtosis: Choose to display the kurtosis value. Calculates kurtosis based on the entire population.
SEM: Choose to display the standard error of the mean.
Sum: Choose to display the data sum.
Minimum: Choose to display the data minimum.
Maximum: Choose to display the data maximum.
Range: Choose to display the data range. Data Range is the difference between the maximum and
minimum.
Quartiles: Choose to display the first quartile, the median and the third quartile.
Interquartile range: Choose to display the difference between the first and third quartiles.
Deciles: Choose to display the nine values that divide the sorted data into ten equal parts.
Percentiles: To request a percentile:
• Enter the desired value in the box placed in Percentiles group box. For example, if you wanted the
7th
percentile, you would enter a 7 in the box.
• Click the add button to add the percentile to the list of requested percentiles.
• Repeat Step 1 and 2 to add additional percentiles as desired.
• If you need to delete a percentile, select it in the list and click the remove button.
Mode: Choose to display the mode and the number of times it occurs. If multiple modes exist, Minitab
displays the smallest modes, up to a total of four, along with their frequency.
Sum of squares: Choose to display the sum of the squared data values. This is the uncorrected sums of
squares, without first subtracting the mean.
MSSD: Choose to display half the Mean of Successive Squared Differences.
N nonmissing: Choose to display the number of nonmissing column entries.
N missing: Choose to display the number of missing column entries.
N total: Choose to display the total (nonmissing and missing) number of column entries.
5

Cut–Points: Divide the data into a number of equal groups. For example, to create deciles, you would
enter 10 in the box .Enter 3 to divide the data into tertiles.
Check statistics
Check None: Choose to clear all check boxes and then individually check the statistics to display.
Check All: Choose to check all boxes. You can uncheck statistics as needed.
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3. – Crosstabs
Dialog box items
Available Variables list shows numeric and text data columns containing at least one no-missing value.
Enter at least one variable in the Rows box. Enter at least one variable in the Columns box.
Cells Dialog box
Check the content of the table cells.
Statistics Dialog box
Check the test to perform.
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The Chi-Square box includes:
• Pearson Chi-Square Test
• Yates's Continuity Correction (only for 2-by-2 square tables)
• Likelihood Ratio G-Test
• Mantel-Haenszel Chi-Square Test
For further details, see Methods and Formulas Help: Pearson Chi Square Test.
The Fisher’s Exact Test box includes:
• One sided and two sided Fisher’s Exact Test (only for 2-by-2 square tables)
For further details, see Methods and Formulas Help: Fisher’s Exact Test.
The McNemar box includes:
If the table is a 2-by-2 square matrix
• McNemar asymptotic
• Edwards Continuity Correction
• McNemar Exact Binomial
• Mid-P McNemar Test
If the table is a k-by-k square matrix, with k > 2
• McNemar-Bowker Test
For further details, see Methods and Formulas Help: McNemar’s Test.
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The Risk box includes:
• Odds Ratio
• Relative Risk
• Attributable risk
• Relative Attributable Risk
• Number Needed to Harm
• Attributable Risk per Unit
• Etiologic Fraction
Risk Estimate statistics are only computed for a 2-by-2 square table without empty cells
For further details, see Methods and Formulas Help: Risk Test.
The Kappa box includes:
• Cohen's Kappa Test (only for k-by-k square tables)
For further details, see Methods and Formulas Help: Cohen's Kappa Test.
The Contingency Phi Cramer box includes:
• Phi Coefficient
• Contingency Coefficient
• Standardized Contingency Coefficient
• Cramer's V
• Tschuprow's T
The Lambda box includes:
• Symmetric Lambda
• Asymmetric Lambda (Row variable as dependent)
• Asymmetric Lambda (Column variable as dependent)
The Goodman and Kruskal tau box includes:
• Asymmetric Uncertainty Coefficient (Row variable as dependent)
• Asymmetric Uncertainty Coefficient (Column variable as dependent)
The Uncertainty Coefficient box includes:
• Symmetric Uncertainty Coefficient
• Asymmetric Uncertainty Coefficient (Row variable as dependent)
• Asymmetric Uncertainty Coefficient (Column variable as dependent)
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The Ordinal box includes:
• Gamma
• Sommers’ d
• Kendall’s tau-b
• Kendall’s tau-c
10

4. – One Sample Test
Overview
In the box Test Type, choose the test to perform:
• Z Test
• T Test
• Variance Test
Hypothesis Test
To perform a hypothesis test, check the box Perform Hypothesis Test and choose the alternative
hypothesis of the test:
• Less than: Perform a level α test of H0: µ ≥ µ0 against the one-sided alternative H1: µ < µ0
• Not Equal: Perform a level α test of H0: µ = µ0 against the two-sided alternative H1: µ ≠ µ0
• Greater than: Perform a level α test of H0: µ ≤ µ0 against the one-sided alternative H1: µ > µ0
Significance level of the test (α) derive from the 1-α value set in the Confidence Level text box.
If you choose a lower-tailed hypothesis test, an upper confidence bound will be constructed. If you
choose an upper-tailed hypothesis test, a lower confidence bound will be constructed.
4.1. – Z Test
Use 1-Sample Z to compute a confidence interval or perform a hypothesis test of the mean when the
standard deviation of the population σ is known. The samples should come from a normal population if n
is low; if however n>30 the distribution of the population does not have to be normal.
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Dialog box items
Data: Choose the origin of the data.
• Samples in columns: Choose if you have entered raw data in columns. Enter the columns
containing the sample data in the list Computing Variables. Move these variables from Available
Variables list to Computing Variables list using Add and Remove buttons. Enter the value for the
population standard deviation in the text box. Entering multiple columns, ISSTATS performs
separate one-sample analyses on each column.
• Summarized data: Choose if you have summary values for the sample size and mean.
o Sample size: Enter the value for the sample size.
o Mean: Enter the value for the sample mean.
o Population Standard deviation: Enter the value for the population standard deviation.
Confidence level: Enter the level of confidence desired. Enter any number between 0 and 1. Entering 0,9
will result in a 90% confidence interval. The default is 0,95 = 95%.
Perform hypothesis test: Check to perform the hypothesis test.
• Hypothesized mean: Enter the test mean µ0.
• Alternative hypothesis: Choose the alternative hypothesis of the test.
4.2. – T Test
Performs a one-sample t-test or t-confidence interval for the mean.
Use T Test for one sample to compute a confidence interval and perform a hypothesis test of the mean
when the population standard deviation, σ, is unknown. Use this test when samples come from a normal
population or n > 30.
Dialog box items
Variables list to Computing Variables list using Add and Remove buttons. Entering multiple
columns, ISSTATS performs separate one-sample analyses on each column
• Summarized data: Choose if you have summary values for the sample size and mean.
o Mean: Enter the value for the sample mean.
o Sample Standard deviation: Enter the value for the sample standard deviation.
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• Hypothesized mean: Enter the test mean µ0.
4.3. – Variance Test
This procedure calculates confidence intervals for the variance of a population, and performs a hypothesis
test to determine whether the population variance equals a specified value. Use this test when samples
come from a normal population.
Dialog box items
Variables list to Computing Variables list using Add and Remove buttons. Entering multiple
columns, ISSTATS performs separate one-sample analyses on each column.
• Summarized data: Choose if you have summary values for the sample size and variance.
o Sample Variance: Enter the value for the sample variance.
• Hypothesized variance: Enter the test variance σ2
0.
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5. – Two-Sample Test
• 2-Sample t-Test
• Paired t-Test
• 2 Variances F-Test
Hypothesis Test
To perform a hypothesis test, check the box Perform Hypothesis Test and choose the Alternative
Hypothesis of the test:
• Less than: Perform a level α test of H0: µ ≥ µd against the one-sided alternative H1: µ < µd
• Not Equal: Perform a level α test of H0: µ = µd against the two-sided alternative H1: µ ≠ µd
• Greater than: Perform a level α test of H0: µ ≤ µd against the one-sided alternative H1: µ > µd
Significance level of the test (α) derive from the 1-α value set in the Confidence Level text box.
If you choose a lower-tailed hypothesis test, an upper confidence bound will be constructed. If you
choose an upper-tailed hypothesis test, a lower confidence bound will be constructed.
Available Variables list shows numerical and text columns containing at least one no-missing value.
Text columns may be used as Subscripts.
5.1. – 2 Sample t-Test
Performs an independent 2-sample t-test and generates a confidence interval.
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When you have dependent samples, use 5.2. – Paired t-Test.
Use 2-Sample t-Test to perform a hypothesis test and compute a confidence interval of the difference
between two population means when the population standard deviations, σ's, are unknown.
Dialog box items
• Samples in one column: Choose if the sample data are in a single column, differentiated by
subscript values (group codes) in a second column.
o Computing Variables: Enter the columns containing the sample data in the list
Computing Variables. Move these variables from Available Variables list to Computing
Variables list using Add / Remove buttons.
o Grouping Variable: Enter the column containing the sample subscripts to define the
groups. It may be a numerical or text column.
o Group 1/Group 2: Enter the subscripts that define both groups.
o Automatic Grouping: Check to let the application define automatically both groups. It
will identify the first two different subscripts in the grouping variable.
• Samples in different columns: Choose if the data of the two samples are in separate columns.
o First Sample: Enter the column containing one sample from Available Variables list.
o Second Sample: Enter the column containing the other sample from Available Variables
list.
• Summarized data: Choose if you have summary values for the sample size, mean, and variance
for each sample.
o First Sample
 Sample size 1: Enter the sample size for the first sample.
 Mean 1: Enter the value for the mean of the first sample.
 Sample Variance 1: Enter the value for the variance of the first sample.
o Second Sample
 Sample size 2: Enter the sample size for the second sample.
 Mean 2: Enter the value for the mean of the second sample.
 Sample Variance 2: Enter the value for the variance of the second sample.
Assume Population equal variances: Check to assume that the populations have equal variances.
• Hypothesized mean: Enter the hypothesized difference between the two population means µ1-µ2.
A difference of 0 suggests in the null hypothesis the equality between mean populations; H0: µ1-µ2 = 0
against an alternative H1: µ1-µ2 ≠ 0
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A difference equal or less than 0 suggest in the null hypothesis that the first mean is equal or less than the
second; H0: µ1-µ2 ≤ 0 against an alternative H1: µ1-µ2 > 0
A difference equal or greater than 0 suggest in the null hypothesis that the first mean is equal or greater
than the second; H0: µ1-µ2 ≥ 0 against an alternative H1: µ1-µ2 < 0
Optionally, test ratios other than 0 (equality) can be specified. A difference of 2 suggests in the null
hypothesis the first mean is the second mean plus 2.
5.2. – Paired t-Test
Performs a paired t-test. This is appropriate for testing the mean difference between paired observations
when the paired differences follow a normal distribution.
Use the Paired t command to compute a confidence interval and perform a hypothesis test of the mean
difference between paired observations in the population. A paired t-test matches responses that are
dependent or related in a pairwise manner. A typical example of the repeated measures t-test would be
where subjects are tested prior to a treatment, say for high blood pressure, and the same subjects are
tested again after treatment with a blood-pressure lowering medication. Paired samples t-tests are often
referred to as "dependent samples t-tests". When the samples are drawn independently from two
populations, use 5.1. – 2 Sample t-Test.
Dialog box items
• Sample in columns: Choose if you have entered raw data in two columns.
o First sample: Enter the column containing the first sample from Available Variables list.
o Second sample: Enter the column containing the second sample from Available Variables
list.
Pairs must have two numerical values. Pairs that have a missing data in any of the
members are ignored.
• Summarized data: Choose if you have summary values for the sample size, mean, and variance
of the difference.
o Mean of Differences: Enter the value for the mean of differences 𝑑𝑑̅.
o Variance of Differences: Enter the value for the variance of differences s2
d.
• Hypothesized mean: Enter the hypothesized population mean of the paired differences µd.
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5.3. – 2 Variances F-Test
The 2 Variances F-Test procedure performs hypothesis tests and computes confidence intervals for the
ratios between two populations' variances. Use this test to determine if one treatment condition has more
variability than the other. Each population must follow the normal distribution.
Dialog box items
• Samples in one column: Choose if you have entered data into a single column with a second
column of subscripts that identify the samples.
o Computing Variables: Enter the columns containing the sample data in the list
Computing Variables. Move these variables from Available Variables list to Computing
Variables list using Add and Remove buttons. Entering multiple columns, ISSTATS
performs separate analyses on each column.
o Grouping Variable: Enter the column containing the sample subscripts to define the
groups. It may be a numerical or text column.
o Group 1/Group 2: Enter the subscripts that define both groups.
o Automatic Grouping: Check to let the application define automatically both groups. It
will identify the first two different subscripts in the grouping variable.
• Samples in different columns: Choose if you have entered the data for the two samples into
separate columns.
o First Sample: Enter the column that contains the data for the first sample from Available
Variables list.
o Second Sample: Enter the column that contains the data for the second sample from
Available Variables list.
• Summarized data: Choose if you have summary values for the sample sizes and variances.
o First Sample
 Sample size 1: Enter the sample size for the first sample.
 Sample Variance 1: Enter the variance for the first sample.
o Second Sample
 Sample size 2: Enter the sample size for the second sample.
 Sample Variance 2: Enter the variance for the second sample.
• Hypothesized ratio: Enter the hypothesized ratio between two population’s variances σ2
1/σ2
2.
A ratio of 1 suggests in the null hypothesis the equality between variance populations; H0: σ2
1/σ2
2 = 1
against an alternative H1: σ2
1/σ2
2 ≠ 1.
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A ratio equal or less than 1 suggest in the null hypothesis that the first variance is equal or less than the
second; H0: σ2
1/σ2
2 ≤ 1 against an alternative H1: σ2
1/σ2
2 > 1.
A ratio equal or greater than 1 suggest in the null hypothesis that the first variance is equal or greater than
the second; H0: σ2
1/σ2
2 ≥ 1 against an alternative H1: σ2
1/σ2
2 < 1.
Optionally, test ratios other than 1 (equality) can be specified. A ratio of 2 suggests the first variance is
double the second variance.
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6. – One-Way ANOVA
Performs a one-way analysis of variance. You can also perform multiple comparisons. The response
variable must be numeric. The factor level column (grouping variable) may be numeric or text.
Dialog box items
• Groups in different Columns
o Computing Variables: Enter the columns containing the response. Each column must
contain the data for one of the groups.
• Groups in 1 Column
o Computing Variables: Enter the column or columns containing the response. ISSTATS
performs a ANOVA test for each of these columns.
o Grouping Variable: Enter the column containing the factor levels.
Confidence level: Enter the confidence level.
For further details, see Methods and Formulas Help: ANOVA Test.
Multiple Comparisons: Use to generate grouping information tables and confidence intervals for the
differences between means, by different methods. Check to obtain confidence intervals for all pairwise
differences between level means using any of these methods.
• Scheffe
• Tukey HSD: Tukey's Honestly Significant Difference test.
• Sidak
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• Fisher LSD: Fisher's Least Significant Difference test.
• Bonferroni
For further details, see Methods and Formulas Help: ANOVA Multiple Comparisons.
Include Brown-Forsythe Test for equality of Means
Use to test the equality of means when distribution violates the assumption of equal variances.
The Brown-Forsythe test cannot be computed if all groups have zero variance. To take part into Brown–
Forsythe test, a group must have at least two elements. In the situation that some groups have zero
standard deviations, the statistic can be computed but the approximation may not work.
For further details, see Methods and Formulas Help: Brown–Forsythe Test for equality of means.
Include Welch’s Test for equality of Means
Use to test the equality of means when distribution violates the assumption of equal variances.
To take part into Welch’s test, a group must have non zero variance. Moreover, sample sizes of a group
have to be greater than or equal to 2.
For further details, see Methods and Formulas Help: Welch’s Test for equality of means.
Remark: ISSTATS does not cancel a test if any of the groups does not fulfill the conditions. It simply
rejects the group. To take part into an ANOVA test, a group must have at least one element. To take part
into a Welch’s Test, a group must have at least two elements; moreover, the group must have a non-zero
variance. Thus, ISSTATS may use 4 of the groups to perform ANOVA Test but only 2 of the groups to
perform a Welch’s Test. It depends on the number of elements of each group. You should read the
Descriptive Statistics information to check the Rejected Groups and also should read the Total Groups
info of each test to check the number of groups being used in that test.
Outputs
Outputs include:
• Descriptive statistic for every level of the factor’s variable.
• Total mean.
• Degrees Of Freedom: DFTotal, DFInter and DFIntra
• The Sum Of The Squares: SSTotal, SSInter and SSIntra
• The Mean Square: MSTotal, MSInter and MSIntra
• The F ratio
• The p-value
• R-squared
• R-squared adjusted
Inter is also referred as Between Groups or Between Treatments. Intra is also referred as Within Groups
or Error Term.
20

R-squared represents the percentage of variation in a response variable that is explained by its relationship
with one predictor variable.
𝑅𝑅2
=
𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆
= 1 −
R-squared adjusted is a version of r-squared that has been adjusted for the number of predictors in the
model. R-squared tends to overestimate the strength of the association, especially when there are more
than one independent variables.
𝑅𝑅2
𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = 1 −
𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀
𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀
Watch One-Way ANOVA Menu Video Help
21

7. – Homoscedasticity Tests
These tests are used to test if k samples are from populations with equal variances. The response variable
has to be numeric. The factor level column (grouping variable) may be numeric or text.
Dialog box items
• Levene's Test
• Brown–Forsythe Test for equality of variances
• Bartlett's Test
For further details, see Methods and Formulas Help: Homoscedasticity Tests.
Remark: ISSTATS does not cancel a test if any of the groups does not fulfill the conditions. It simply
rejects the group. To take part into a Levene or Brown–Forsythe test, a group must have at least one
element. To take part into a Bartlett's Test, a group must have at least two elements; moreover, the group
must have a non-zero variance. Thus, ISSTATS may use 4 of the groups to perform Levene's Test but
only 2 of the groups to perform a Bartlett's Test. It depends on the number of elements of each group.
You should read the Descriptive Statistics information to check the Rejected Groups and also should read
the Total Groups info of each test to check the number of groups being used in that test.
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8. – Bivariate Correlation Tests
Dialog box items
• Covariance. Computes the matrix of covariances.
• Pearson. Computes the Matrix of the Pearson Product-Moment Correlation Coefficients.
• Tau-b Kendall. Computes the Matrix of Kendall's Tau-b Correlation Coefficients.
• Spearman. Computes the Matrix of Spearman’s Correlation Coefficients.
Missing Values. You can choose one of the following:
• Exclude cases pairwise. Cases with missing values for one or both of a pair of variables for a
correlation coefficient are excluded from the analysis. Since each coefficient is based on all cases
that have valid codes on that particular pair of variables, the maximum information available is
used in every calculation. This can result in a set of coefficients based on a varying number of
cases.
• Exclude cases listwise. Cases with missing values for any variable are excluded from all
correlations.
Outputs
Outputs include:
• Coefficient. It may be the Pearson, Spearman or Tau-b Kendall coefficient.
• p-value (Bi.). A two-tailed significance level (bilateral test).
• Significant. “YES” if the test is statistically significant (p-value <= α = 1 – Confidence Level).
“NO” if the test is NOT statistically significant (p-value > α = 1 – Confidence Level).
• N. Number of cases or paired observations (Xi, Yi) in the sample.
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For further details, see Methods and Formulas Help: Bivariate Correlation Tests.
Remark: All tests use pairwise deletion mode, thus the statistical procedure uses cases that contain some
missing data. A case may contain 3 variables: VAR1, VAR2, and VAR3. A case may have a missing
value for VAR1, but this does not prevent some statistical procedures from using the same case to analyze
variables VAR2 and VAR3. Pairwise deletion allows you to use more of your data. However, each
computed statistic may be based on a different subset of cases.
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9. – Parametric Value at Risk
From the menus choose: Analyze > Forecast > Parametric Value at Risk
Compute the Parametric Value at Risk by the Variance-Covariance Method.
Inputs
Choose the columns for daily return rates of each asset in a portfolio. Each of these columns must have a
list of return rates for an asset on a daily basis. Do not enter the return rates in percent values (%); use per
unit values.
Once you add the columns, input the position on each asset. Set the Holding Period (trading days) and the
Confidence Level (1-α).
Outputs
• Descriptive statistics of each asset daily return
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• For each asset, Position, One-day Expected Return, Standard Deviation or Volatility, One-day
Value at Risk, Total Value at Risk for n Trading Days, One-day Expected Shortfall.
• Covariance matrix
• One-day Portfolio Expected Return
• Portfolio Variance
• Portfolio Standard Deviation or Portfolio Volatility
• One-day Value at Risk
• Holding Period in Trading Days
• Total Value at Risk for n Trading Days
• One-day Portfolio Expected Shortfall
• For each asset, Marginal Value at Risk, Component Value at Risk, Incremental Value at Risk.
Remark: There is no a common definition of VaR. Sometimes VaR is assumed to be the Portfolio
Volatility as expected return is supposed to be zero. ISSTATS follows the method described on J.P.
Morgan webpage, thus it does NOT consider VaR as Portfolio Volatility. Please, check the equations at
Methods and Formulas Help: Parametric Value at Risk.
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10. – Exponentially Weighted Moving Average (EWMA)
Forecast
From the menus choose: Analyze > Forecast > EWMA Forecast
ISSTAS follows the J.P. Morgan RiskMetrics© approach to estimating and forecasting volatility by
exponentially weighted moving average model (EWMA).
Inputs
Choose the columns for daily return rates of each asset in a portfolio. Each of these columns must have a
list of Continuously Compounded Return for an asset on a daily basis. Do not enter the return rates in
percent values (%); use per unit values.
Order is the time order of the data series:
• Ascending: the first data point –the value in the first row- corresponds to the earliest date.
• Descending: the first data point – the value in the first row- corresponds to the latest or most
recent date.
Lambda is the smoothing parameter or decay factor used for the exponential-weighting scheme. A default
value of 0.94 is shown.
Set the forecast time/horizon (t). If you set 1 as forecast time, application computes the forecast for the
next day, given the information up to and including the last day of the series.
Outputs
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• Descriptive statistics of each asset daily return
• For each asset, Position, One-day Expected Return, Volatility by EWMA method, One-day Value
at Risk, Total Value at Risk for n Trading Days.
• One-day Covariance matrix by EWMA method.
• Correlation matrix by EWMA method.
• One-day Portfolio Expected Return
• Portfolio Variance.
• Portfolio Volatility.
• One-day Value at Risk
• Holding Period in Trading Days
• Total Value at Risk for n Trading Days
• For each asset, Marginal Value at Risk, Component Value at Risk, Incremental Value at Risk.
Remarks
• The time series is supposed homogeneous or equally spaced. Missing values are removed from the
series.
• The daily returns should be computed as Continuously Compounded Return: ln(Si/Si-1).
• The time series should have more than 50 values.
• When computing EWMA volatility, ISSTATS assumes that the time series has an average equal
to zero; but it does not assumes the average is zero when computing the VaR (VaR = μ + zασ).
• ISSTATS does not use the recursive formula for EWMA; hence it does not compute any initial
variance.
Please, check the method and assumptions at Methods and Formulas Help: Exponentially Weighted
Moving Average (EWMA) Forecast.
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11. – Financial Formulas
From the menus choose: Analyze > Financial Formulas
1. Select the variables required for the formula and move them into the fields. Each formula have
its own requirements. The fields are: Daily High Price, Daily Low Price, Daily Close Price,
Daily Volume and Daily Price.
2. Select the formula you want to apply from the drop-down list. Once selected, the fields that
requires the formula will be colored in a paled color.
3. Set the parameters for the formula.
See below the requirements for each formula.
Accumulation Distribution Formula
The accumulation distribution formula calculates a cumulative total of prices and volumes.
An increase in the accumulation distribution index indicates a probable price increase. A decrease in the
accumulation distribution index indicates a probable price decrease.
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Input Values
This formula takes four input variables.
• High: Daily high price.
• Low: Daily low price.
• Close: Daily close price.
• Volume: Daily volume.
Output Value
This formula outputs one variable.
• AD: Accumulation distribution index.
Average True Range Formula
The average true range formula records the maximum values of the following three differences, and
calculates the moving average of the resulting data series:
• Between the previous day's high and low prices.
• Between the previous day's close price and the current day's high price.
• Between the previous day's close price and the current day's low price.
The average true range indicator is a good measure of commitment. A high value often indicates market
bottom due to panic sell. A low value often indicates market top.
Parameters
This formula takes one optional parameter.
• Period: Period for calculating the moving average of the true range values. The default value is 14.
Input Values
This formula takes one input variable.
30

Output Value
• ATR: Average true range indicator.
Bollinger Bands Formula
The Bollinger Bands formula calculates the standard deviation above and below a simple moving average
of the data. Since standard deviation is a measure of volatility, a large standard deviation indicates a
volatile market, and a smaller standard deviation indicates a calmer market.
Parameters
This formula takes two required parameters.
• Period: Period for calculating the moving average for the Bollinger Bands.
• StdDev: The number of standard deviations for calculating the upper and lower bands.
Input Values
• Price: The price for which the Bollinger Bands are calculated.
Output Value
This formula outputs two variables.
• Upper Band: Upper Bollinger Band.
• Lower Band: Lower Bollinger Band.
Chaikin Oscillator Formula
The Chaikin Oscillator formula is useful for monitoring volume flow in a market. It applies the
Accumulation Distribution Formula on the input, calculates the exponential moving average of the result
for a short period and a long period, and then outputs the difference between the two.
31

This formula should be used together with the Envelopes Formula.
Parameters
This formula takes two optional parameters.
• Period Short: Period for calculating the short period exponential moving average. The default
value is 3.
• Period Long: Period for calculating the long period exponential moving average. The default
value is 10.
Input Values
Output Value
• CO: Chaikin Oscillator.
Commodity Channel Index Formula
The commodity channel index formula calculates the mean deviation of the daily average price from the
moving average. A value above 100 indicates that the commodity is overbought, and a value below -100
indicates that the commodity is oversold.
Parameters
This formula takes one optional parameter.
• Period: Period for calculating the commodity channel index. The default value is 10.
Input Values
This formula takes three input variables.
32

Output Value
• CCI: Commodity channel index.
Detrended Price Oscillator Formula
The detrended price oscillator formula calculates the difference between the daily price and the moving
average. This is useful for identifying cycles and overbought and oversold price levels.
Parameters
This formula takes one required parameter.
• Period: Period for calculating the moving average for the detrended price oscillator index.
Input Values
• Price: Price for which the detrended price oscillator index is calculated.
Output Value
• DPOI: Detrended price oscillator index.
Ease of Movement Formula
The ease of movement formula uses the close price and volume to measure the strength of the price trend.
A value close to zero indicates that prices are not moving easily, while a high positive value indicates that
prices are going up easily and a high negative value indicates that prices are going down easily.
33

Input Values
Output Value
• EOM: Ease of movement indicator.
Envelopes Formula
The envelopes formula calculates "envelopes" above and below a moving average using a specified
percentage as the shift. The envelopes indicator is used to create signals for buying and selling. You can
specify the percentage the formula uses to calculate the envelopes.
Parameters
This formula takes two required parameters.
• Period: Period for calculating the moving average.
• Shift: Percentage used to shift the upper and lower envelopes from the moving.
Input Values
• Price: Price for which the envelopes are calculated.
Output Value
• Upper: Upper envelope.
• Lower: Lower envelope.
Forecasting Formula
34

The forecasting formula attempts to fit the historical data to a regression function and forecast future
values of the data best on the best fit.
Parameters
This formula takes four optional parameters.
• Regression Type: Indicate one of these regression types: Constant, Linear, Quadratic, Cubic,
Quartic, Exponential, Logarithmic, Power.
• Period: Forecasting period. The formula predicts data for this period of days into the future. The
default value is half of the series' length.
• Approx. Error: Whether to output the approximation error. If unchecked, output error series
contain no data for the corresponding historical data.
• Forecast. Error: Whether to output the forecasting error. If unchecked, output error series contain
the approximation error for all predicted data points if ApproxError is checked.
Input Values
• Price: Historical data of price for forecasting.
Output Value
This formula outputs three variables.
• Forecast: Forecasted values.
• Upper Error: Upper bound error.
• Lower Error: Lower bound error.
Mass Index Formula
The mass index formula predicts trend reversals by calculating the range between high and low prices for
each period. A bulge in the index line signals a possible trend reversal. You can use a 9-day Exponential
Moving Average Formula to determine whether the bulge is a buy or sell signal.
Parameters
This formula takes two parameters.
35

• Period A: Period of accumulation. The default value is 25.
• Period EMA: Period for calculating the exponential moving average for the mass index. The
default value is 9.
Input Values
This formula takes two input variables.
Output Value
• MI: Mass index.
Money Flow Formula
The money flow formula compares upward changes and downward changes of volume-weighted typical
prices. It can be used to identify market tops and bottoms.
Parameters
• Period: Period for calculating the money flow index.
Input Values
Output Value
36

• MF: Money flow index.
Moving Average Convergence/Divergence Formula
The moving average convergence/divergence (MACD) formula compares a short period moving average
and a long period moving average of prices. MACD is used with a 9-day exponential moving average as a
signal that identifies buying or selling moments.
Parameters
This formula takes two optional parameters.
• Short Period: Period for calculating the short period moving average. The default value is 12.
• Long Period: Period for calculating the long period moving average. The default value is 26.
Input Values
• Price: The price to be calculated by the formula.
Output Value
• MACDIndicator: Distance between the short and long exponential moving averages.
Exponential Moving Average Formula
The exponential moving average formula is a moving average of data that gives more weight to the more
recent data in the period and less weight to the older data in the period. This formula produces a moving
average that follows the market trend much more quickly than the Weighted Moving Average Formula.
This formula smoothes a data series. This makes analyzing volatile data easier.
Parameters
37

• Period: Period for calculating the exponential moving average.
Input Values
• Price: Price for which the exponential moving average is calculated.
Output Value
• EMA: Exponential moving average.
Simple Moving Average Formula
The simple moving average formula takes the average of data over a period of time, and "moves" the
period across the data series one data point at a time. This formula smoothes a data series and makes
analyzing volatile data easier.
Parameters
• Period: Period for calculating the moving average.
Input Values
• Price: Price for which the moving average is calculated.
Output Value
• SMA: The moving average.
Triangular Moving Average Formula
38

The triangular moving average formula takes a simple moving average of data and applies a simple
moving average on the first moving average. It is a lagging indicator, and is always behind the price. The
triangular moving average gives the most weight to the middle portion of the data.
Parameters
• Period: Period for calculating the moving averages.
Input Values
• Price: Price for which the triangular moving average is calculated.
Output Value
• TMA: Triangular moving average.
Triple Exponential Moving Average Formula
The triple exponential moving average formula is useful for eliminating short and insignificant cycles in
the data. It smoothes the data three times using the Exponential Moving Average Formula, and then
calculates the rate of change in the moving average based on the result for the previous day.
Parameters
• Period: Period for calculating the exponential moving average for the triple exponential moving
average indicator.
Input Values
• Price: Price for which the triple exponential moving average indicator is calculated.
39

Output Value
• TEMA: Triple exponential moving average.
Weighted Moving Average Formula
The weighted moving average formula is a moving average of data that gives more weight to the more
recent data in the period and less weight to the older data in the period. This formula smoothes a data
series. This makes analyzing volatile data easier.
Parameters
• Period: Period for calculating the weighted moving average.
Input Values
• Price: Price for which the weighted moving average is calculated.
Output Value
• WMA: Weighted moving average.
Negative Volume Index Formula
The negative volume index formula helps identify a bull market. When the negative volume index is
above its moving average there is higher probability for a bull market. The probability for a bull market is
much lower when the negative volume index is below its moving average.
This formula should be used together with the Positive Volume Index Formula.
40

Parameters
• Start NVI: Start value of the negative volume index.
Input Values
Output Value
• NVI: Negative volume index.
On Balance Volume Formula
The on balance volume formula measures positive and negative volume flows.
Input Values
Output Value
• OBV: On balance volume indicator.
Performance Formula
41

The performance formula calculates the rate of price change compared with historical data. It differs from
the Rate of Change Formula in that it calculates rate of change against the first available data. The output
is a percentage.
This formula can also be used to calculate the rate of volume change.
Input Values
• Price: Price for which the performance indicator is calculated. It can be any other data for which
you wish to calculate the performance indicator, such as volume.
Output Value
• Performance: Performance indicator.
Positive Volume Index Formula
The positive volume index formula helps identify a bear market. When the positive volume index is
below its moving average there is higher probability for a bear market. The probability for a bear market
is much lower when the positive volume index is above its moving average.
This formula should be used together with the Negative Volume Index Formula.
Parameters
• Start PVI: Start value of the positive volume index.
Input Values
42

Output Value
• PVI: Positive volume index.
Median Price Formula
The median price formula calculates the average value of daily high and low prices.
Input Values
Output Value
• MP: Median price indicator.
Typical Price Formula
The typical price formula calculates the average value of daily high, low, and close prices.
Input Values
Output Value
43

• TP: Typical price indicator.
Weighted Close Formula
The weighted close formula calculates the average value of daily prices, but gives more weight to the
close price.
Input Values
Output Value
• WC: Weighted close indicator.
Price Volume Trend Formula
The price volume trend formula calculates a cumulative volume total using relative changes of the close
price. A bullish divergence between the price volume trend indicator and the price indicates that the
market is at the bottom. A bearish divergence between the price volume trend indicator and the price
indicates that the market is at the top.
Input Values
Output Value
44

• PVT: Price volume trend indicator.
Rate of Change Formula
The rate of change formula calculates the rate of price change compared with historical data. It differs
from the Performance Formula in that it calculates the rate of change against a period of days prior to the
current price. The output is a percentage.
This formula can also be used to calculate the rate of volume change.
Parameters
This formula takes one parameter.
• Period: Number of days prior to the current day. The formula uses the data from that day as a
reference when calculating the current rate of change. The default value is 10.
Input Values
• Price: Price for which the rate of change is calculated. It can be any other data for which you wish
to calculate the rate of change, such as volume.
Output Value
• Rate: Rate of change indicator.
Relative Strength Index Formula
The relative strength index formula is a momentum oscillator formula that compares upward movements
of the close price with downward movements, and outputs values from 0 to 100. A value close to 100
indicates that the price is about to move downward, and a value close to 0 indicates that the price is about
to move upward.
45

Parameters
• Period: Period for calculating the relative strength index. The default value is 10.
Input Values
• Price: The closing price for which the relative strength index is calculated.
Output Value
• RSI: Relative strength index.
Standard Deviation Formula
The standard deviation formula is used to indicate volatility. It calculates the difference between values
like the close price and their moving average. A higher standard deviation indicates higher volatility.
Parameters
• Period: Period for calculating the moving average for the standard deviation.
Input Values
• Price: Price for which the standard deviation is calculated.
Output Value
• StdDev: Standard deviation.
46

Stochastic Indicator Formula
The stochastic indicator formula calculates the simple stochastic indicator (%K) and the smoothed
stochastic indicator (%D). %D is a moving average of %K. The output is a percentage. A value more than
80% indicates that the current price is close to the price high, and a value less than 20% indicates that the
current price is close to the price low.
The stochastic indicator is an indicator of upward and downward market trends.
Parameters
• Period K: Period for calculating %K. The default value is 10.
• Period D: Period for calculating %D. The default value is 10.
Input Values
Output Value
• %K: Simple stochastic indicator. The default value is 10.
• %D: Smoothed stochastic indicator. The default value is 10.
Volatility Chaikins Formula
The Volatility Chaikins formula calculates the exponential moving average of the difference between
daily high and low prices, then calculates the rate of change of the exponential moving average.
Parameters
47

• Period EMA: Period for calculating the exponential moving average of the difference between the
high and low prices. The default value is 10.
• Period ROC: Period for calculating the rate of change. The default value is 10.
Input Values
Output Value
• VC: Volatility Chaikins indicator.
Volume Oscillator Formula
The volume oscillator formula measures the difference between a short period moving average of volume
and a long period moving average of volume. A positive value indicates a strong trend, and a negative
value indicates a weak trend.
Parameters
This formula takes three parameters.
• Period Short: Period for calculating the short period moving average. The default value is 5.
• Period Long: Period for calculating the long period moving average. The default value is 10.
• Use Percentage: Whether to output the difference in percentage. When set to false, the formula
outputs the difference as a point. The default value is true.
Input Values
• Volume: Volume for which the volume oscillator indicator is calculated.
Output Value
48

• VO: Volume oscillator indicator.
William's %R Formula
The William's %R formula is a momentum indicator, and is used to measure over-bought or oversold
levels. This indicator is very similar to the stochastic %K indicator, except that the Williams %R formula
calculates a negative value between 0 and -100 and does not smooth the data.
Parameters
• Period: Period for calculating the Williams %R indicator. The default value is 14.
Input Values
• High: The high price.
• Low: The low price.
• Close: The closing price.
Output Value
• WilliamsR: Williams %R indicator.
Watch Financial Formulas Menu Video Help
49

12. – Linear Regression
From the menus choose: Analyze > Linear Regression
1. Select a numeric dependent variable.
2. Select one or more numeric independent variables.
Data. The dependent and independent variables should be quantitative. Categorical variables need to be
recoded to binary (dummy) variables or other types of contrast variables.
Settings Tab
Confidence Interval. Enter a value between 0,0001 and 0,9999 to specify the confidence level for the
ANOVA test and Prediction Intervals. Entering 0,9 will result in a 90% confidence interval. The default is
0,95 = 95%.
Include constant in equation. By default, the regression model includes a constant term. Deselecting this
option forces regression through the origin, which is rarely done. Some results of regression through the
origin are not comparable to results of regression that do include a constant. For example, R 2 cannot be
interpreted in the usual way.
Output Descriptive. Provides the number of valid cases, the mean, and the standard deviation for each
variable in the analysis.
50

Output Covariance Matrix for Coefficients. Displays a variance-covariance matrix of regression
coefficients with covariances off the diagonal and variances on the diagonal.
Output Durbin Watson statistic. Displays the Durbin-Watson statistic.
Save Tab
You can save predicted values, residuals, and other statistics useful for diagnostic information. Each
selection adds one or more new variables to your active data file.
• Unstandardized Predicted Values. The value the model predicts for the dependent variable.
• Prediction Intervals for Mean. Lower and upper bounds (two variables) for the prediction interval
of the mean predicted response.
• Prediction Intervals for Individual. Lower and upper bounds (two variables) for the prediction
interval of the dependent variable for a single case.
• Standardized Predicted Values. A transformation of each predicted value into its standardized
form. That is, the mean predicted value is subtracted from the predicted value, and the difference
is divided by the standard deviation of the predicted values. Standardized predicted values have a
mean of 0 and a standard deviation of 1.
• Mahalanobis Distances. A measure of how much a case's values on the independent variables
differ from the average of all cases. A large Mahalanobis distance identifies a case as having
extreme values on one or more of the independent variables.
• Cook's Distances. A measure of how much the residuals of all cases would change if a particular
case were excluded from the calculation of the regression coefficients. A large Cook's D indicates
that excluding a case from computation of the regression statistics changes the coefficients
substantially.
• Centered Leverage values. Measures the influence of a point on the fit of the regression. The
centered leverage ranges from 0 (no influence on the fit) to (N-1)/N.
• Unstandardized Residuals. The difference between an observed value and the value predicted by
the model.
• Standardized Residuals. The residual divided by an estimate of its standard deviation.
Standardized residuals, which are also known as Pearson residuals, have a mean of 0 and a
standard deviation of 1.
• Studentized Residuals. The residual divided by an estimate of its standard deviation that varies
from case to case, depending on the distance of each case's values on the independent variables
from the means of the independent variables.
To get predicted values for cases that were not used in the regression analysis, provide rows with values
in the predictor variables and missing values in the response variable.
51

52

13. – Curve Estimation
From the menus choose: Analyze > Curve Estimation
1. Select one or more numeric dependent variables.
2. Select one or more numeric independent variables.
Data. The dependent and independent variables should be numeric. A separate model is produced for
each pair of dependent-independent variables.
You can choose one or more curve estimation regression models. Available regression models are:
• Linear
• Quadratic
• Cubic
• Quartic
• Quintic
53

• Sextic
• Inverse
• Logarithmic
• Power
• Exponential
• Compound
• S-curve
• Logistic
• Growth
Include constant in equation. Estimates a constant term in the regression equation. The constant is
included by default.
Plot models. Plots the values of the dependent variable and each selected model against the independent
variable. A separate chart is produced for each dependent variable. Optionally, select a numeric or a string
variable in Label Cases By field, for labeling cases in scatterplots. You can label points on the plot with
this variable.
a. If selected, the value labels (or values if no labels are defined) of this variable are used as
point labels.
b. If you do not select a variable to label cases by, other options can be used to label points.
i. No Label
ii. Case Number
iii. Coordinates
iv. Y value
Check to save to save predicted values, residuals, and prediction intervals as new variables. Set the
Confidence Interval for prediction intervals.
54

14. – Create Time Series
From the menus choose: Analyze > Forecast > Create Time Series
1. Select the time series function that you want to use to transform the original variable(s).
2. Set the parameters for the function.
The Time Series Transformation Functions are:
Difference. Nonseasonal difference between successive values in the series. The order is the number of
previous values used to calculate the difference.
Seasonal difference. Difference between series values a constant span apart. The span is the periodicity.
The order is the number of seasonal periods used to compute the difference.
Centered moving average. Average of a span of series values surrounding and including the current value.
The span is the number of series values used to compute the average.
Prior moving average. Average of the span of series values preceding the current value. The span is the
number of preceding series values used to compute the average.
55

Running medians. Median of a span of series values surrounding and including the current value. The
span is the number of series values used to compute the median.
Cumulative sum. Cumulative sum of series values up to and including the current value.
Lag. Value of a previous case, based on the specified lag order. The order is the number of cases prior to
the current case from which the value is obtained.
Lead. Value of a subsequent case, based on the specified lead order. The order is the number of cases after
the current case from which the value is obtained.
56

15. – Univariate GARCH(1,1)
From the menus choose: Analyze > Forecast > Univariate GARCH(1,1)
Available Variables list shows numeric data columns containing at least one no-missing value. Enter at
least one variable in the Computing Variables box.
The time series is supposed homogeneous or equally spaced. Missing values are removed from the series.
The model specification is
• x(t) = μ + ε(t)
• ε(t) = z(t)·σ(t), where z(t) ↝ N(0, 1), iid
• σ2
(t) = ω + α ·ԑ2
(t-1) + β ·σ2
(t-1)
57

Settings Tab
Maximization Algorithm.
ISSTATS uses the Maximum Likelihood (MLE) estimation method to get the parameter estimates of the
model. The log-likelihood function of the GARCH(1,1) model with normal distribution becomes
L(μ, ω, α, β) = -1/2·∑ [ln(2π) + α·ln(σ2
t) + β·ϵ2
t/ σ2
t]
ISSTAS has two iterative algorithms for maximizing the above function.
Differential evolution. Differential evolution (DE) is a metaheuristics optimization algorithm. It
is slow, but stable and accurate.
Random BFGS. Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm is an iterative method for
solving bounded nonlinear optimization problems. It is fast but instable. Because of the instability,
ISSTATS performs several BFGS using random initial points (μ0, ω0, α0, β0), and gets the best result. You
can set the Number of BFGS optimization process to perform. Default value is 20. You should test
Random BFGS a couple of times; if it does not converge, set a higher value for Number of BFGS
optimization process to perform.
You should test both algorithms; study the results and always choose the output with lower Log-
likelihood; that is, the maximum absolute value of Log-likelihood.
Presample Values Box.
58

Specifies the initial or priming values for ε2
(t) and σ2
(t) series; that is, ε2
(0) and σ2
(0). It is assumed that
ε2
(0) = σ2
(0).
OLS Unconditional Variance. The OLS estimate of the unconditional variance is ∑(Xi - μ)2
/n.
On each iteration, it’s computed using the current estimate of μ in that iteration. So, the series is
initialized in the j-th iteration as ε2
(0) = σ2
(0) = ∑(Xi - μj)2
/n.
Expected Unconditional Variance. On each iteration, it is computed using the current parameter
estimates as E[σ2
] = ω/(1 – α – β). So, the series is initialized in the j-th iteration as ε2
(0) = σ2
(0) = ωj/(1 –
αj – βj).
Direct Input [ε2(0) σ2(0)]. You can enter two values separated by a white space. The first value
would be ε2
(0); second value would be σ2
(0).
Predict Box.
Mean. The predicted values for Xi series is always the estimated parameter μ. It is not time
varying.
CI for Mean; cond. mean error. The confidence interval for the mean values, using the
conditional mean errors. The mean error are the predicted conditional standard deviations.
CI for Mean; uncond. mean error. The confidence interval for the mean values, using the
unconditional mean errors. The mean error is the square root of the unconditional standard deviation,
computed as ∑(Xi - μ)2
/n. It is not time varying.
Conditional Std. Deviation. It is the square root of the predicted Conditional Variance.
Conditional Variance. The predicted conditional variance is computed this way
• σ2
(t+1) = ω + α·ϵ2
(t) + β·σ2
(t), where ϵ2
(t) and σ2
(t) are the last fitted values of the series.
• σ2
(t+2) and subsequent values are computed recursively as σ2
(t+i) = ω + α· σ2
(t+i-1) +
β·σ2
(t+i-1) = ω + [α + β]·σ2
(t+i-1)
Forecast time/horizon. It’s an integer value, denoting the number of steps to be forecasted, by
default 1.
Confidence Level. It’s used to compute the confidence intervals mentioned above.
Output
The GARCH(1,1) Parameters Output shows the parameter estimation and the robust standard error of the
estimation. The robust standard errors are due to quasi maximum likelihood estimation (QMLE) as
opposed to (the regular) maximum likelihood estimation (MLE). They are robust against violations of the
distributional assumption, e.g. when the assumed distribution is Normal. Bollerslev and Wooldridge
(1992) proved that the QMLE estimates are consistent and asymptotically normally distributed.
59

Save Tab
You can save predicted values, residuals, and other statistics useful for diagnostic information. Each
selection adds one or more new variables to your active data file.
Fitted Values (μ). The fitted values for Xi series is always the estimated parameter μ. It is not
time varying.
Conditional Variance. It is computed as σ2
(t) = ω + α·ϵ2
(t-1) + β·σ2
(t−1)
Conditional Std. Deviations (Volatility). The Conditional Standard Deviations –also known as
Volatility- are the square root of the conditional variance.
Residuals. The residuals are computed as Xi – μ.
60

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