univariate box‐jenkins forecasts of water discharge in missouri river

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Univariate box‐jenkins forecasts of water discharge inMissouri riverRamu Govindasamy aa Research Assistant, Department of Economics , Iowa State University , Ames, IA, 50011,USAPublished online: 02 May 2007.

To cite this article: Ramu Govindasamy (1991) Univariate box‐jenkins forecasts of water discharge in Missouri river,International Journal of Water Resources Development, 7:3, 168-177, DOI: 10.1080/07900629108722509

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Univariate Box—Jenkinsforecasts of water discharge inMissouri river

Ramu Govindasamy

The Missouri river is one of the major rivers in the USA and is used for variouspurposes such as irrigation, electricity generation, fishing and other recreationalactivities. Predicting the flow of water would help to plan all activities that aredependent on this river. The results of Univariate Box—Jenkins forecasts show thatthe Seasonal AutoRegressive Moving Average (SARIMA) model with logarithmictransformation of the form SARIMA (0, 1, 1) (0, 1, 1)12 is a fairly good parsi-monious representation of the underlying process.

Introduction

Irrigation water is a scarce resource and an importantinput to agriculture. The importance of irrigationwater lies not only in its own productivity but also inits ability to increase the productivity of other cropproduction inputs such as fertilizer (Eswaramoorthyet al, 1989). Thus there is an imperative need for theefficient use of this scarce resource, to increaseagricultural production to meet the demand for foodfor an increasing population (Govindasamy andBalasubramanian, 1990 and Govindasamy andPalanisami, 1990). Since many rivers are used forirrigation, predicting the flow of water in the riverwould help in agricultural production. The fanningcommunity can plan: what crop to grow, when togrow and the timing of operation of each practice.The Missouri is one of the most important rivers inthe USA used for the purpose of irrigation. Landsurveys in Missouri showed that flood-plain forestsoccupied 76% of the land area in 1826 and 13% in1972. Cultivated land increased from 18% to 83% inthe same time period (Hesse et al, 1988) and 80% ofthe flood plain was under cultivation by 1958 (Braggand Tatschl, 1977). The area under agriculture in1982 was 100 091 ha. Predicting the flow of water in

Ramu Govindasamy is Research Assistant, Department ofEconomics, Iowa State University, Ames, IA 50011, USA.

this river would also help in estimating the energygeneration, the probable time of flooding, etc. Thereare many ways by which a series can be forecasted.One such method is to extrapolate the past patternsinto the future, using the univariate Box-JenkinsARIMA (AutoRegressive Integrated Moving Aver-age) process. This type of forecasting is more scien-tific and gives reasonable forecasts of the series.Bessler (1980) used ARIMA models for estimatingthe yield of field crops.

The objective of this article is to consider differentmodels for water discharge (Z,) forecast in theMissouri river and compare the results and the pointforecasts of the selected models. The next sectiondescribes the Univariate Box—Jenkins modellingprocedure. The source and nature of data followed byrange—mean analysis are then discussed. Identifica-tion of different models based on the preliminaryestimates are discussed, followed by the modelling,estimation and diagnostic checking and point fore-casts and forecast intervals. The final section givesthe conclusions of the results.

The univariate Box—Jenkins (UBJ)modelling procedure

The UBJ modelling procedure has three stages. Stageone deals with tentative identification of the models.

168 0790-0627/91/030168-10 © 1991 Butterworth-Heinemann Ltd

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Forecasts of water discharge in Missouri river: Ramu Govindasamy

Figure 1. Time plot of water discharge (ft /sec).

In stage two, the model parameters are estimated andin stage three diagnostic checking is carried out. Themodel can be identified using various techniques suchas time plot of the data, autocorrelation functions(ACF), partial autocorrelation functions (PACF), etc.The estimation can be carried out using non-linearleast squares and diagnostic checking can be doneusing residual analysis. If the model satisfies thediagnostic checking conditions, the analysis is com-plete; otherwise the three stages should be carried outagain until the model passes the diagnostic checking.UBJ modelling is used in cases of short-termforecasting when the univariate time series dataavailable are equally spaced and stationary.1 Non-stationary processes can be transformed into station-ary processes by differencing. A sample with morethan 50 observations is normally required for theanalysis. The advantages of UBJ modelling are that:

• it is based on statistical theory;• it has an optimality property for forecasts, which

means that no other univariate forecasts have asmaller mean-squared forecast error;

• it is a flexible class of models;• it can be generalized to handle transfer functions,

intervention, and multivariate problems;• the model is obtained from the data.

Source and nature of data

The data were collected from the US GeologicalSurvey water data report 'Water Resources Data -Water Year 1988', which is an annual publicationstarted in 1964. The series, mean monthly values ofwater discharge in the Missouri river, was measuredat Nebraska City, Nebraska. The series has 297 meanmonthly observations, measured in cubic feet persecond.

'Z, is a stationary process if £(Z,) = / i ; , is a constant, variance(Z,) = a\ is also a constant and cov(Z,, Z, + j) = n, which dependsonly o n t .

As a first step in identifying a parsimonious (Pan-cratz, 1983) model for estimation and forecasting, theseries was plotted against time. Figure 1 shows 297mean monthly observations of water discharge in theMissouri river plotted against time, with time on thehorizontal axis and water discharge on the verticalaxis. As can be seen from the graph, the mean of theprocess remains almost the same but the variance ofthe process changes slightly over time.

Range-mean analysisThe range-mean plot is used to test whether a seriesis stationary or not. This can be done as a four-stepprocedure:

(1) divide the realization into groups;(2) transform with chosen r, where r is defined as in

(l);(3) compute the mean and range in each group; and(4) plot the range V mean.

This procedure should be repeated for differentvalues of r, until the plot does not show any trend.For the range-mean plot, the series was divided into27 groups, with 11 observations in each group. The

90000

80000

70000

60000

c 50000S.

40000

30000

20 000

10 000'20000 30000 40000 50000 70000 70000

Mean

Figure 2. Range-mean plot of water discharge in Missouririver.

WATER RESOURCES DEVELOPMENT Volume 7 Number 3 September 1991 169

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range of the mean of each group is plotted againsirespective mean. The non-homogeneous (Box

theand

Jenkins, 1976) nature of the data can be seen from therange-mean plot of the series as shown in Figure 2,if the outliers are discarded. To identify the correctmodel both logarithmic transformation and no trans-formation were used in estimation. Thereforemodels with following transformation,

Z*= (Z, + my

the

(1)

where r = 1, 0 and m = 0, were compared forforecasting. The time plot of the natural logarithm ofwater discharge is shown in Figure 3 and the cor-responding range-mean plot is given in Figure 4 Ascan be seen from the plot, after logarithmic transfor-mation the range-mean plot does not show a linearrelationship, which implies that the process is station-ary (Priestly, 1974).

The modellingThe principle behind UBJ—ARIMA modelling isthe observations in a time series may be statisticallyrelated to other observations in the same series. JTheUBJ-ARIMA model helps to find a good way ofstating that statistical relationship (Pancratz, 1983).An ARIMA model is an algebraic statement showinghow a time series variable (Z,) is related to its ownpast values (Z,_,, Z,_2, Z,_3,mon ARIMA processes are

Z, = C — 0| a, _ i + a,

.). The two com-

(2)

(3)

Equation (2) shows how Z, is related to its ownimmediate past value (Z,_|), which is called anautoregressive (AR) process. The longest time lagassociated with the Z term on the right-hand side iscalled the AR order of the process. Equation (2) is anAR(1), because the longest time lag attached to a pastZ value is one period. C is a constant term. (/>, is a

1.8

1.6

1.4

1.2

g, 1.0co01 0.8

0.60.4

0.2

•

-

+

-

X

• 0 oH

-

_

10.0 10.2 10.4 10.6 10.8 II.0 11.2Mean

Figure 4. Range-mean plot of natural log of water dischargein Missouri river.

fixed coefficient whose value is determined by therelationship between Z, and Z, _,. The a, term is aprobabilistic 'shock' element. The terms C, 0|Z,_,and a, are each components of Z,. C is a deter-ministic, 0!Z,_, is a probabilistic since its valuedepends in part on the value of Z,_,, and a, is apurely probabilistic component.

Now consider equation (3). Processes with pastrandom shocks are called moving average (MA) pro-cesses. The longest time lag is called the MA orderof the process. Equation (3) is an MA process oforder one (MA[1]), since the longest time lagattached to a past random shock is t — 1. The negativesign attached to 0, is merely a convention. The MAprocesses are also univariate, because past random-shock terms can be replaced by past Z terms throughalgebraic manipulations.

Time-series data are often periodic in behaviour. Aperiodic series has a pattern which repeats every stime periods, where s > 1. ARIMA models are oftenbest suited for forecasting periodic data series (Pan-cratz, 1983). The most common periodic behaviour is

120 144 168 192 216 240 264 288

Figure 3. Time plot of natural

170

log of water discharge — seasonally adjusted.

WATER RESOURCES DEVELOPMENT Volume 7 Number 3 September 1991

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seasonal variation. The letter s stands for the lengthof periodicity. SARIMA is the seasonal autoregres-sive integrated moving average model built using thesame iterative modelling procedure used for non-seasonal data: identification, estimation and diagno-stic checking. The seasonal time-series data mustoften be differenced by length s to attain stationarity.

The two common seasonal processes are theseasonal autoregressive process and seasonal movingaverage process, which can be expressed as follows.

(4)

(5)~ C-Qsa,_s

Equation (4) says that Z, is related to its own pastvalue s periods earlier, Z,_,. Similarly, in equation(5), 2, is related to the random shock s periodsearlier, a,-s. The estimated autocorrelation functions(ACF) and partial autocorrelation functions (PACF)must be compared with theoretical ACFs and PACFsfor the purpose of identification.

The estimated ACF and the PACF of the series Z,for lags = 36 is shown in Figure 5. It indicates thatthe ACF dies down with a wave-like pattern withhighly significant spikes at lags, 12, 24, 36, . . .indicating a strong seasonal pattern. The ACFindicates that the period of seasonal pattern, S couldbe 12. The estimated PACF cuts off after one spike.Since there is a strong seasonality in period 12, aseasonal difference of order one was taken. Figure 6shows the ACF and PACF of seasonally differenced

- I

ACF

" nm. " [ • •

12 16 20 24 28 32 36Lag

PACF

t : : I T : 1 - : : - - : -11 ill I I I I_ I I i ii:_:_il I I I I _ I I i

0 4 8 12 16 20 24 28 32 36Lag

Figure 6. Estimated ACF and PACF of seasonally adjustedseries.

ACF

: ITTTTT—i T ; t l t : Ttt

0 4 8 12 16 20 24 28 32 36Lag

' PACF

- I

•

u

_ T _ TT TT

I 1

" T 7

1

- f

0 4 8 12 16 20 24 28 32 36Lag

Figure 5. Estimated ACF and PACF of original series.

series. As can be seen from the graph, the ACF diesdown linearly, but not very slowly and the PACF cutsoff after one lag. This suggests that a regular dif-ference of order one might be helpful. Therefore aregular differencing of order one, on the seasonal dif-ferencing, was carried out. Since, for some, the ACFmight look good enough, not calling for a regular dif-ferencing, a model with no regular differencing wasalso estimated. Figure 7 shows the ACF and PACFof the regular differencing over the seasonally dif-ferenced series. The PACF indicates that the lags at12, 24, 36, . . . die down seasonally and the ACFcuts off at 12. This suggests including a seasonalmoving average term of order 12. For the regularterm, since the ACF cuts off after one lag and thePACF dies down, a moving average (MA) of orderone was included in the model.

Five models were identified at each of the threestages, where stage 1 deals with no transformation,one seasonal differencing (Newbold and Reed, 1979),one regular differencing, as suggested by Figure 6. Instage 1, all five models have the common term


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12 16 20 24 28 32 36

Figure 7. Estimated ACF and PACF of differenced andseasonally adjusted series.

(1 - B) (1 - B12) Z, on the left-hand side, represen-ting a regular differencing and a seasonal differenc-ing. The notation used in SARIMA modelling has ageneral form of

SARIMA (p, d, q) (P, D, Q)s where

p is the order of autoregressiond is the number of regular differencingq is the order of moving averageP is the order of seasonal autoregressionD is the number of seasonal differencingQ is the order of seasonal moving averages is the length of the season

Stage 2 deals with logarithmic transformation, oneregular differencing and one seasonal differencing assuggested by the range-mean plot in Figure 2. Stage3 deals with no transformation, no regular differenc-ing and one seasonal differencing as suggested by tneplot of the time series in Figure 1. Thus, the tentativemodel identification is suggestive of the following 15seasonal autoregression models (SARIMA):

172

Stage 1

Model 1.1: SARIMA (0,1,1) (0,l,l)12

- B I2)Z, = (1 -

Model 1.2: SARIMA (0,1,2) (0,l,l)12

= (l-6lB-02B2Kl-9nBll)a,

Model 1.3: SARIMA (2,1,0) (0,l,l) l2

(1 - 4>,B - <A2S2)(1 - B){\ - Bn)Z,

Model 1.4: SARIMA (1,1,1) (0,l,l)12

{\ - 4>XB){\ - B){\ - Bn)Z,= (l-0]B-d2B

2)(l-enBn)a,

Model 1.5: SARIMA (2,1,1) (0,l,l) l2

(1 - frB ~ 4>2B2){\ - B){\ - Bl2)Z,

Stage 2

Model 2.1: SARIMA (0,1,1) (0,l,l)12

(1 -B)(\ -5 l 2 ) lnZ , = (l - 0 , 5 ) 0 ~QnBl2)

Model 2.2: SARIMA (0,1,2) (0,l,l)12

-fi l 2)InZ,

Model 2.3: SARIMA (2,1,0) (0,l,l) l2

(1 - 0 ,0 - 4>2B2)(l -B)(l -Bx2)lnZ,

= (1 - e,2B12)a,

Model 2.4: SARIMA (1,1,1) (0,1, l) l 2

( l-^BXl -B)(l -BI2)lnZ,

= (l-6lB-62B2K\-GnBl2)a,

Model 2.5: SARIMA (2,1,1) (0,l,l)12

(1 - <f>tB - <f,2B2)(l -B)(l - B 1 2 ) l n Z ,

= (1 - 0,fl)U - ei2fl12)a,

Stage 3

Model 3.1: SARIMA (0,1,1) (0,l,l) l2

Model 3.2: SARIMA (0,1,2) (0,l,l)12

(1 - BI2)Z, = (1 - 6,B - 62B2){\ - QnBn)a,


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Model 3.3: SARIMA (2,1,0) (0,l , l)1 2

(1 - 4>,B - <j>2B2)(l - B12)Z, = (1 - e

Model 3.4: SARIMA (1,1,1) (0,1,1 )12

= (1 - dtB - 62B2){\ - Ql2B

l2)a,

Model 3.5: SARIMA (2,1,1) (0,l,l)I2

(I - W - <t>2B2)(l - B]2)Z,

where

<£, and <f>2 are the regular AR coefficients.0, and 62 are the regular MA coefficients.0 ] 2 is the seasonal MA coefficient.B is the lag operator.a, is the white noise disturbance term,a, ~ N(0, o2

u).

Estimation and diagnostic checking

The preliminary identification has led to fiveSARIMA models in each of the three stages men-tioned in the previous section. These models wereestimated by a conditional least-squares criterion.The coefficient estimates and the other statistics forthe five models in stages 1, 2 and 3 are given inTables 1, 2 and 3 respectively. Table 1 shows that themodel 1.2 has residual ACF significant spike at lagthree and the other models have residual ACF signifi-cant spike at many lags except model 1.1 (Box and

Pierce, 1970). Box—Pierce statistics seems to begood for models 1.1 and 1.2. Two variables weresignificant for models 1.1 and 1.2, which wereselected for comparing the forecasts, in stage 1.These two models are also parsimonious and have asmall standard deviation, when compared with othermodels (Granger and Newbold, 1975).

In stage 2 model 2.5 was non-stationary. Model 2.5had one non-significant coefficient. In selecting twomodels in stage 2 to compare the forecasts, model 2.3was rejected because it had a higher standard devia-tion. Model 2.4 was selected because it had aminimum standard deviation and model 2.1 wasselected based on the parsimonious principle. In stage3, models 3.1 and 3.2 had many ACF significantspikes and model 3.5 was non-stationary. Thereforemodels 3.3 and 3.4 were selected from stage 3, forcomparing the ex-post forecasts. These two modelsalso had the minimum standard deviation.

Forecasts and forecast intervals

Table 4 compares the ex-post forecasts of six selectedmodels, two from each of the three stages. Thesecond column shows the actual flow from period 121to 144. The comparison between the point forecastsfrom six selected models shows that model 1.1 andmodel 1.2 are fairly close to the actual flows whencompared with the other models. In the other models,even though some values are closer to the actualflows, many point forecasts are too high. Thereforemodels 1.1 and 1.2 were selected for forecasting andprediction interval as shown in Table 5. In comparingmodels 1.1 and 1.2 for ex-post point forecast

Table 1. Comparison of

Coefficients/statistics

Parameters

AR(1) (<*>,)

AR(2) (02)

MA(1) (0,)

MA(2) (fl2)

SMA(12) (0I2)

Standard deviation

Residual statistics

ACF significant spikesBox-P ie rce x 2

five alternative models —

Model t . l

0.3835(6.95)

0.9286(26.38)

7.89

Clean16.153

stage 1.

Model 1.2

0.3657(6.16)0.0951

(1.60)0.9300

(26.74)7.87

315.466

Model 1.3

-0.3224(-5.44) '-0.1180

(-1-99)

0.9323(27.03)

7.95

3, 2419.340

Model 1.4

0.5758(7.68)

0.8857(20.43)

0.9303(26.50)

7.84

2, 4, 2417.300

Model 1.5

-0.3245(-0.16)-0.0379

(-0.07)-0.0277

(-0.01)

0.9296(26.53)

7.99

2, 3, 22, 2424.886

Notes: 't-ratios are in parentheses.(rf= 1, D= 1, and r = 1).


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Table 2. Comparison of five alternative models - stage 2.

Coefficients/statistics

ParametersAR(1) («i)

AR(2) (<fe)

MA(1) ($,)

MA(2) (02)

SMA(12) (9,j)

Standard deviation

Residual statistics

ACF significant spikesSox—Pierce x2

Model 2.1

0.3374(5.99)

0.9197(26.65)

0.1617

213.480

M<>del 2.2

0.3219(5.41)0.1030

(1.75)0.9195

(26I.64)0.1612

Clean12.1284

Model 2.3

-0.2891(-4.89)'-0.1287

(-2.19)

0.9225(26.86)

0.1622

313.922

Model 2.4

0.6625(10.98)

0.9233(31.24)

0.9193(26.20)

0.1606

414.814

Model 2.5

-1.1861

-0.1946(-37.37)-0.9875

(-1.93)

0.8060(1.32)0.1713

1,223.560

Notes: '/-ratios are in parentheses.{d=\,D = 1 , and r = 0).

Table 3. Comparison of five alternative models — stage 3.

Coefficients/statisticsParametersAR(1) (<t>,)

AR(2) (<fc)

MA(1) (0,)

MA(2) 0i)

SMA(12) (9,2)

Standard deviation

Residual statistics

ACF significant spikes

Box —Pierce x2

Notes: "/-ratios are in parentheses.(d = 1, D = 1, and r = 1).

Model 3.1

-0.5100(-10.08)

0.9286(26.56)

8.98

1.2, 3,4, 5171.311

Model 3.2

0.621711.41)

- 0.3960(-J7.27)

[0.9287(26.24)8.17

3,6,65

, 5,24226

Model 3.3

0.6002(10.23)'

0.1713(2.92)

0.9300(26.26)

7.61

24

11.714

Model 3.4

0.8392(18.83)

0.2494(3.21)

0.9301(26.37)

7.61

24

10.943

Model 3.5

0.8975(2.85)

-0.0442(-0.19)

0.3057(1.00)

0.9301(26.32)

7.62

24

14.759

estimates, the values are closer to the actual flow inthe case of model 1.1 and model 1.1 is parsimonious.Figure 8 shows the normal probability plot of theresiduals for model 1.1. Since it looks fairly linear,the normality assumptions of the residuals hold(Durbin, 1960). The residual ACF as shownFigure 9 also looks clean, which implies thatmodel is a good fit. Figure 10 shows the plot of

bythethe

residuals v time which is not correlated betweenperiods. Figure 11 shows the pot of the residuals vforecast values for model 1.1. They seem to be evenly

174

distributed around the mean. Plot of the 95% predic-tion interval for model 1.1 is given in Figure 12.

Application of the model for better watermanagement

Even though the model is applied to forecast thewater discharge in the Missouri river, it can bemodified to forecast the flow in any river which hassome qualities of this river. When the water dischargehas a seasonal variation, forecasting the flow of water


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Table 4.

Period

121122123124125126127128129130131132133134135136137138139140141142143144

Comparison of forecasts

Actual flow"

26.21035.38036.64044.49044.84040.35036.39037.76036.81038.06036.65023.01020.76022.10031.17044.67044.23053.15056.98065.54066.51066.18068.48038.060

for six alternative

Model 1.1

19.95626.321.38.67647.35345.58949.99542.74141.96242.78245.19743.52625.99119.90226.26738.62247.29945.53549.94142.68741.63842.72745.14343.47225.937

models.

Model 1.2

20.15026.16438.55247.27345.43049.78242.54640.99642.12244.57442.87825.41819.31025.60037.98846.70944.86649.21841.98240.43241.55844.01042.31424.854

Model 2.1

20.57527.28539.35248.57747.73751.94445.25444.14144.87846.99045.20726.84121.00927.86040.18149.60248.74353.039

• 46.20845.07245.82447.98046.16027.407

Model 2.4

21.80229.49343.10353.71953.21058.31151.16750.35051.21553.62251.56330.56323.95431.84145.99656.88556.06061.22753.60552.67253.52456.00253.82831.897

Model 3.3

20.59827.49040.25149.35547.86052.56845.71145.09745.73547.85046.18428.72422.45228.88341.40550.28648.61753.18246.20945.50146.06348.11646.40028.899

Model 3.4

20.65127.34240.08749.14047.60152.25345.32844.60745.18947.70746.05328.60422.38828.80041.31050.16748.46252.97645.93445.11645.61648.06646.35328.856

Note: "Mean monthly water discharge in Missouri river (in 1000 ft'/scc).

Table 5.

Period

298299300301302303304305306307308309310311312313314315316317318319320321

Forecast of mean monthly

Forecast

39.15738.24822.51415.95021.21834.33343.23341.61143.64036.64034.36735.22637.10636.19720.46313.89919.16732.28241.18239.56041.84234.58932.31633.175

water discharge in Missouri

Model 1.1

95% limits

Lower

23.68920.077

1.9920.0000.0007.990

15.21712.01612.8004.1170.4750.0180.3250.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

river."

Upper

54.62556.42043.03638.57945.77560.67671.24971.20674.98669.16268.25970.43473.88774.37159.98154.71661.24475.58285.67185.20788.61982.46881.27483.188

Forecast

39.02637.68121.94015.37020.63133.74342.64540.99743.28936.04933.62834.49436.37735.42219.68113.11118.37231.48440.38638.73841.02933.79031.36932.235

Model 1.2

Lower

23.60019.4141.8680.0000.0009.035

16.57513.63214.6876.2622.7012.4683.0010.8520.0000.0000.0000.0000.4960.0000.0000.0000.0000.000

95% limits

Upper

54.45155.94942.01337.09843.89758.45168.71668.36371.89165.83664.55566.52069.75469.99255.37949.90256.36870.36880.27779.61082.85976.55675.05176.815

Note: "Water discharge is measured in 1000 ftVsec):


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Forecasts

3

2

1

0

-1

- 2

-3

-

-

-

-

-

-

-

-

*

of water discharge

/

/

/

/

*

in Missouri river: Ramu C

/ -f

1 i

dvindasamy

50000

40000

30000

20000| 10000'm _a> 0

a:-10000

-20000

-30000-40000

ic

-40000-20000 20000 40 000 60000Residuals

Figure 8. Residual probability plot for Model 1.1.

- I

'TT

0 4 8 12 16 20 24 28 32 36Log

Figure 9. Residual autocorrelation function for Model

40

.1.

would help to decide in advance the crops to begrown. If the model predicts a decreased water flowin the river, the cropping pattern can be modifiedaccording to the available water for irrigation. In theagricultural sector, when most of the decisions madeare irreversible, forecasting the water discharge helps

• . * t

:t$O&*$tt>;:-. \ •

10000 30000 50000 70000 90000 110000Forecasted values

Figure 11. Plot of residual v forecast values for Model 1.1.

85920|

71891

0 57862

1 43833U

^ 29804!£ 15775oS 1747

-12284

-26312297 309 32!

Time

Figure 12. Plot of 95% prediction interval for Model 1.1.

to choose an optimal plan increasing the net profits ofa farm. For example, if a producer decides to growcorn, once the seeds are shown and when the corn isabout two months old, if irrigation water is notavailable during the third month, the producer willlose the crop. This is, of course, on the assumptionthat there is no alternative source of irrigation water.At the same time, in the presence of forecastedvalues, the producer could have chosen a crop whichrequired less irrigation water or could have planted adrought-resistant variety. Since most farming opera-

24 48 72 96

Figure 10. Plot of residual v time for Model 1.1.

176

120 144 168 192 216 240 264 288


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tions are interlinked, predicting the flow would helpto improve the efficiency of more than one operation.Prediction also helps to improve the efficiency ofother activities such as the energy generation, rivercleaning projects and fishing. The forecasts can alsobe used to judge the probable time of flooding of theriver.

ConclusionThe above results indicate that the model 1.1,

SARIMA (0,1,1) (0, l , l ) l 2

- Bn)Z, = (1 -

is a fairly good parsimonious representation. Itexplains the underlying data-generating mechanismmore closely than the other fourteen models. In viewof this the water discharge in Missouri river fore-casts, based on model 1.1, will be more precise andcan trace the movements in the series closely. Anyriver which has the characteristics of the Missouririver, but not necessarily the same level of discharge,can also be predicted for water flow.

ReferencesBessler, D.A. (1980). 'Aggregated personalistic beliefs on

yield of selected crops estimated using ARIMA process',American Journal of Agricultural Economics, Vol 62, No 4,pp 666—674.

Box, G.E.P. and G.M. Jenkins (1976). Time series:Forecasting and Control, rev. edn. Holden Day, San Fran-cisco.

Box, G.E.P. and D.A. Pierce (1970). 'Distribution of residualautocorrelations in autoregressive-integrated moving

average time series models', Journal of the AmericanStatistical Association, Vol 64, pp 1509—1526.

Bragg, T.B. and A.K. Tatschl (1977). 'Changes in flood plainvegetation and land use along the Missouri river from 1826to 1972', Environmental Management, Vol 1, No 4, pp343-348.

Durbin, J. (1960). 'The fitting of time series models', Reviewof the International Institute of Statistics, Vol 28, pp233-240.

Eswaramoorthy, K., R. Govindasamy and I. Singh (1989).'Integrated use of water resources in the lower Bhavani pro-jects in India', Water Resources Development, Vol 5, No 4,pp 279-286.

Govindasamy, R. and R. Balasubramanian (1990). 'Tankirrigation in India: problems and prospects', WaterResources Development, Vol 6, No 3, pp 211—217.

Govindasamy, R. and K. Palanisami (1990). 'Optimal moder-nization for a tank irrigation system using a simulationmodel', Indian Journal of Agricultural Economics, Vol 45,pp 141-149.

Granger, C.W.J. and P. Newbold (1975). 'The time-seriesapproach to econometric model building', paper presentedto the Seminar on New Methods in Business CycleResearch, Federal Reserve Bank of Minneapolis, 13-14November.

Hesse, W.H., C.W. Wolfe and N.K. Cole (1988). 'Someaspects of energy flow in the Missouri River ecosystem anda rationale for recovery', in Norman G. Benson, ed. TheMissouri River, The Resources, Their Uses and Values,American Fishery Society, Special Publication No 8.

Newbold, P. and G.V. Reed (1979). 'The implications foreconomic forecasting of time series model buildingmethods, in O.D. Anderson, ed. Forecasting, NorthHolland, New York.

Pancratz, A. (1983). Forecasting With UnivariateBox—Jenkins Models — Concepts and Cases, John Wiley,New York.

Priestly, M.B. (1974). Comments on the paper 'Experiencewith forecasting univariate time series and the combinationof forecasts', by P. Newbold and C.W.J. Granger, Journalof the Royal Statistical Society Series A., Vol 137, pp131-165.

Water Resources Data, Iowa, Water Years 1964-1988, USGeological Survey Water Data Report IA-88-1.


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univariate box‐jenkins forecasts of water discharge in missouri river

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