## May 8, 2012

### PM0015 [Quantitative Methods in Project Management] Set1 Q3

Q.3. Describe the time series forecasting with the help of autoregressive modeling. What do you understand by managing cash flow?

Autoregressive modeling is another approach in forecasting with the annual time-series data. Frequently, the values of a series of data at particular points in time are highly correlated with the values that precede and succeed them. A first-order autocorrelation refers to the magnitude of the relationship between the values that are two periods apart. A p-th order autocorrelation refers to the size of the correlation between values in a time-series that are p periods apart. To obtain a better historical fit of the data and, at the same time, make useful forecasts of their future behaviour, it is possible to take advantage of the autocorrelation inherent in such data by considering autoregressive modelling methods.

After the model is selected and the least-squares regression methods are used to obtain estimates of the parameters, the next step is to determine the appropriateness of this model. Either it is possible to select a particular p-th order autoregressive model based on previous experiences with similar data, or as a starting point, it is also possible to choose a model with several parameters and then to eliminate the parameters that do not contribute significantly. In the latter approach, a t-test for the significance of Ap, the highest order autoregressive parameter in the current model under consideration, is used. The null and the alternative hypothesis are as follows –
H0: Ap = 0
H1: Ap ≠ 0

The test statistic of t-Test for Significance of the Highest-order Autoregressive Parameter Ap is the following -
pappSAat
Where
Ap = hypothesized value of the highest-order parameter Ap in the regression model
ap = the estimate of the highest-order parameter Ap in the regression model
= the standard deviation of ap paS

And the test statistic t follows a t-distribution having n-2p-1 degrees of freedom.

The following equations describe a set of autoregressive models –
First Order Autoregressive Model

Second Order Autoregressive Model

p-th Order Autoregressive Model

where
Yi = the observed value of the series at time i
Yi-1 = the observed value of the series at time i-1
Yi-2 = the observed value of the series at time i-2
Yi-p = the observed value of the series at time i-p
A0 = fixed parameter to be estimated from least-squares regression analysis
A1, A2, …., Ap = Autoregression parameters to be estimated from the least-squares regression analysis

= a non-auto-correlated random (error) component (with mean = 0 and constant variance)

The first-order autoregressive model is similar in the form to the linear regression model, the second-order autoregressive model is similar to the multiple regression model with two independent variables and the p-th order autoregressive model, is similar in the form to the multiple regression model. In the regression models, the regression parameters are given by the symbols , , , ….., , with corresponding estimates denoted by b0, b1, …., bk. In the autoregressive models, the parameters are given by the symbols A0, A1, …. , Ap, with the corresponding estimates denoted by a0, a1, ….., ap.

A first order autoregressive model is concerned only with the correlation between consecutive values in a series. A second-order autoregressive model considers the effects of the relationship between the consecutive values in a series and between values that are two periods apart. A p-th order autoregressive model deals with the effects of the relationships between consecutive values, values two periods apart, and so on – up to the values p periods apart.

The selection of an appropriate autoregressive model is a complex task. It is needed to weigh the advantages that are due to simplicity against the concern of failing to take into account important autocorrelation behaviour inherent in data. On the other hand, it is needed to be concerned with the selection of a higher-order model requiring the estimation of numerous, unnecessary parameters – specially if n, the number of observations in the series, is not too large. The reasons for this concern is that p out of n data values will be lost in obtaining an estimate of Ap when comparing each data value with another data value, which is p periods apart.

Managing Cash Flow
Managing cash flow in a project is an important task to be performed. Managing cash flow involves making sure that sufficient payments are received from the customer in time so that one has enough money to cover the costs of performing the project – employee payroll, charges for materials, invoices from subcontractors, and travel expenses, for example.

The key to managing cash flow is to ensure that cash comes in faster than it goes out. If sufficient cash isn’t available to meet expenses, money must be borrowed. Borrowing increases project cost because any money borrowed must be paid back to the lender, along with a charge for borrowing the money – the interest. The flow of cash coming in from the customer can be controlled by the terms of payment in the contract. From the contractor’s point of view, it’s desirable to receive payments from the customer early in the project rather than later.

The contractor might try to negotiate payment terms that require the customer to do one or more of the following:

1. Provide a down payment at the start of the project. This requirement is reasonable when the contractor needs to purchase a significant amount of materials and supplies during the early stages of the project.
2. Make equal monthly payments based on the expected duration of the project. Cash outflow usually is smaller in the early stages of a project. If more cash is coming in than is going out during the early part of the project, the contractor may be able to invest some of the excess cash and earn interest. The saved funds can then be withdrawn to meet the greater cash outflow requirements later in the project.
3. Provide frequent payments, such as weekly or monthly payments rather than quarterly payments.

The worst scenario from the contractor’s point of view is to have the customer make only one payment at the end of the project. In this situation, the contractor will need to borrow money to have cash available to meet expenses throughout the project.